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: 
Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).

Answer: 
No. This task does not involve an equation in two variables. There is only 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: 
Construct a tangent line from a point outside a given circle to the circle.

Answer: 
No. This task does not involve drawing lines that are tangent to circles. The only line that may be drawn is the ray between points P and B.

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_neg_0

HS_1

im_task_001160_neg_0

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standard_description: Display numerical data in plots on a number line, including dot plots, histograms, and box plots.
problem_activity: Activity
The mathematical purpose of this activity is for students to describe how two variables are related, and to determine whether or not there is a
causal relationship
. Students should begin to recognize that some variables may be related, but one does not cause the other to change. At this point, the mathematics of scatter plot analysis cannot decide whether there is a causal relationship. The relationship must be thought through carefully to decide, based on the situation, whether the related variables have a causal relationship. As students analyze the relationships, they are modeling with mathematics (MP4).
Launch
Arrange students in groups of 2. Tell students there are many possible answers for the questions. After quiet work time, ask students to compare their responses to their partner’s and decide if they are both correct, even if they are different. Follow with a whole-class discussion.
Representation: Develop Language and Symbols.
Create a display of important terms and vocabulary. After the first question invite students to suggest language or diagrams to include that will support their understanding of causal relationships. Examples to include can be phrases, such as, “increased rain causes people to wear jackets.” The two variables can be color-coded in two different colors to reveal the relationship. Both the axes and the labels can be highlighted with the color that connects it to the type of variable it addresses.
Supports accessibility for: Conceptual processing; Language
Student Facing
Each of the scatter plots show a strong relationship. Write a sentence or two describing how you think the variables are related.
[IMG]
During the month of April, Elena keeps track of the number of inches of rain recorded for the day and the percentage of people who come to school with rain jackets.
[IMG]
A school book club has a list of 100 books for its members to read. They keep track of the number of pages in the books the members read from the list and the amount of time it took to read the book.
[IMG]
Number of tickets left for holiday parties at a venue and noise level at the party.
[IMG]
The height and score on a test of vocabulary for several children ages 6 to 13.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Activity Synthesis
The goal is for students to develop an understanding of what it means for a relationship between two variables to be a causal relationship.
Select several students to share their reasoning for the relationship between the variables. For each pair of variables, ask students what might have
caused
the variables to be linked. In some cases (the first two here), one of the variables causes a change in the other. In other cases (the last two here), an additional variable or situation is the cause of the change.
Tell students that, since most people are used to seeing independent variables on the
\(x\)
-axis and dependent variables on the
\(y\)
-axis, the convention is to put the causal variable on the
\(x\)
-axis and the other variable on the
\(y\)
-axis when one of the variables does cause the other to change. The book club scatter plot should probably have the axes switched to meet the convention.
Here are some questions for discussion.
“Why does an increase in precipitation cause an increase in the percentage of people wearing rain jackets?” (When it rains, people usually wear coats to stay dry.)
“An increase in the time it takes to read a book does not cause the number of pages in the book to increase. Does an increase in the number of pages in a book cause the time it takes to read a book to increase? Explain your reasoning.” (Yes, if you switch the axes on the graph, you can see that relationship. It is causal because it takes longer to read more pages.)
“The relationship in the height and test score graphs appears to be strong. Does an increase in height cause an increase in test scores?” (It is not the height that causes the increase in test score, but probably the age or grade level of the students might be causing the increase. Age and grade level are positively related to height, so that likely explains the relationship seen between the two variables.)


aligns: no