few_shots: 
Example 1:
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.

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


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:
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).

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


Answer: 
No. This task does not involve an equation in two variables. There is only one variable, x.

Example 3:
Concept/skill: 
Construct a tangent line from a point outside a given circle to the circle.

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 $P$.


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, determine whether the following problem or activity can enable students to understand the concepts or skills in the provided description.

shot_info: fl_problem_001894_neg_0

HS_0

im_task_000870_neg_0

HS_1

im_task_001160_neg_0

HS_2
standard_description: Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.
problem_activity: Warm-up
The mathematical purpose of this activity is to explore, formally define, and introduce notation for the concept of
conditional probability.
Launch
Display the image of a deck of cards.
[IMG]
Explain that there are four suits (spades, hearts, diamonds, clubs) with 13 cards in each suit (ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, jack, queen, king) for a total of 52 cards. Hearts and diamonds are always considered red cards while clubs and spades are always considered black cards. Face cards are jack, queen, or king.
Unless otherwise noted, a standard deck of cards in these materials will consist of these 52 cards.
Representation: Internalize Comprehension.
Use virtual or concrete manipulatives to connect symbols to concrete objects or values. Provide students with a standard deck of cards to see the different suits and the cards that are red and black. Allow groups of students to manipulate the cards to illustrate the probabilities in the problems.
Supports accessibility for: Visual-spatial processing; Conceptual processing
Student Facing
Noah will select 1 card at random from a standard deck of cards. Find the probabilities. Explain or show your reasoning.
[IMG]
\(P(\text{the card is a queen})\)
\(P(\text{the card is a heart})\)
\(P(\text{the card is a queen and heart})\)
Elena pulls out only the hearts from the deck and sets the rest of the cards aside. She shuffles the hearts and draws one card. What is the probability she gets a queen?
Student Response
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for free access to Student Response.
Activity Synthesis
The purpose of this discussion is to formally define conditional probability and to introduce notation for conditional probability. Tell students that “
conditional probability
is defined as the probability of an event happening under the condition that another event occurs.” Use Elena’s situation as an example to show the notation for conditional probability:
\(P(\text{the card is a queen | the card is a heart})\)
which can be read as “the probability that the card is a queen under the condition that the card is a heart has occurred.” In this situation, Elena made sure the card would be a heart by removing the other suits and drawing only from those cards.
Display
\(P(\text{the card is a heart | the card is a queen})\)
for all to see and ask students to explain what it means, then ask students to find the probability. (It means the probability that the card is a heart under the condition that it is a queen. It has the value of
\(\frac{1}{4}\)
since there is one heart card among the 4 queens.) This is the notation we will use throughout this unit to represent conditional probability. Ask students “How would you write what is the probability that a card is an 8 under the condition that is a spade?” (
\(P(\text{the card is an 8| the card is a spade})\)
).


aligns: no