few_shots: 
Example 1:
Concept/skill: 
Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

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


Answer: 
Yes. 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.

Example 2:
Concept/skill: 
Solve quadratic equations in one variable.

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


Answer: 
Yes. This task shows one quadratic equation with 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_pos_0

HS_0

im_task_000870_pos_0

HS_1

im_task_001160_neg_0

HS_2
standard_description: Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.
problem_activity: Activity
In this activity, students continue to compute probabilities for multi-step experiments using the number of outcomes in the sample space. The first problem involves a situation for which students have already seen the sample space. Following this problem, the class will discuss the merits of the different representations for writing out the sample space. The next two problems involve situations in which students may need to write out the sample space on their own. Students are also reminded that some events have a probability of 0, which represents an event that is impossible. In the discussion following the activity, students are asked to think about the probabilities of two events that make up the entire sample space and have no outcomes common to both events (MP2).
Launch
Keep students in groups of 2.
Assign each group a representation for writing out the sample space: a tree, a table, or a list. Tell students that they should write out the sample space for the first problem using the representation they were assigned. (This was done for them in a previous lesson and they are allowed to use those as a guide if they wish.)
Tell students that they should work on the first problem only and then pause for a discussion before proceeding to the next problems.
Give students 2 minutes of partner work time for the first problem followed by a pause for a whole-class discussion centered around the different representations for sample space.
After all groups have completed the first question, select at least one group for each representation and have them explain how they arrived at their answer. As the groups explain, display the appropriate representations for all to see. Ask each of the groups how they counted the number of outcomes in the sample space as well as the number of outcomes in the event using their representation.
List:
Heads 1, heads 2, heads 3, heads 4, heads 5, heads 6, tails 1, tails 2, tails 3, tails 4, tails 5, tails 6
Table:
| 1  | 2  | 3  | 4  | 5  | 6  |
|----|----|----|----|----|----|
| H1 | H2 | H3 | H4 | H5 | H6 |
| T1 | T2 | T3 | T4 | T5 | T6 |
Tree:
[IMG]
After students have had a chance to explain how they used the representations, ask students to give some pros and cons for using each of the representations. For example, the list method may be easy to write out and interpret, but could be very long and is not the easiest method for keeping track of which outcomes have been written and which still need to be included.
Allow the groups to continue with the remaining problems, telling them they may use any method they choose to work with the sample space for these problems. Give students 10 minutes of partner work time followed by a whole-class discussion about the activity as a whole.
Student Facing
The other day you looked at a list, a table, and a tree that showed the sample space for rolling a number cube and flipping a coin.
Your teacher will assign you one of these three structures to use to answer these questions. Be prepared to explain your reasoning.
What is the probability of getting tails and a 6?
What is the probability of getting heads and an odd number?
Pause here so your teacher can review your work.
Suppose you roll two number cubes. What is the probability of getting:
Both cubes showing the same number?
Exactly
one cube showing an even number?
At least
one cube showing an even number?
Two values that have a sum of 8?
Two values that have a sum of 13?
Jada flips three quarters. What is the probability that all three will land showing the same side?
Student Response
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for free access to Student Response.
Anticipated Misconceptions
Some students may not recognize that rolling a 2 then a 3 is different from rolling a 3 then a 2. Ask students to imagine the number cubes are different colors to help see that there are actually 2 different ways to get these results.
Similarly, some students may think that HHT counts the same as HTH and THH. Ask the student to think about the coins being flipped one at a time rather than all tossed at once. Drawing an entire tree and seeing all the branches may further help.
Activity Synthesis
The purpose of the discussion is for students to explain their methods for solving the problems and to discuss how writing out the sample space aided in their solutions.
Poll the class on how they computed the number of outcomes in the sample space and the number of outcomes in the event for the second set of questions given these options: List, Table, Tree, Computed Outcomes Without Writing Them All Out, Another Method.
Consider these questions for discussion:
“Which representation did you use for each of the problems?”
“Do you think you will always try to use the same representation, or can you think of situations when one representation might be better than another?”
“Did you have a method for finding the number of outcomes in the sample space or event that was more efficient than just counting them?” (The number of outcomes in the sample space for the number cubes could be found using
\(6 \boldsymbol{\cdot} 6 = 36\)
. To find the number of outcomes with at least 1 even number, I knew there would be 6 for each time an even was rolled first and only 3 for each time an odd number was rolled first, so I found the number of outcomes by
\(3 \boldsymbol{\cdot} 6 + 3 \boldsymbol{\cdot} 3 = 27\)
.)
“One of the events had a probability of zero. What does this mean?”
(It is impossible.)
“What would be the probability of an event that was certain?” (1)
“Jada was concerned with having all the coins show the same side. What would be the probability of having at least 1 coin
not
match the others?” (
\(\frac{6}{8}\)
, since there are 6 outcomes where at least 1 coin does not match: HHT, TTH, HTH, THT, HTT, THH.)
“How do the answers to Jada’s question and the one we just answered relate to one another?” (Since every outcome in the sample space has either “at least one heads” or “all tails,” and there is no outcome that applies to both events, together the sum of their probabilities must be 100% or 1.)
Speaking: MLR8 Discussion Supports.
Use this routine to support whole-class discussion. For each response or observation that is shared, ask students to restate and/or revoice what they heard using precise mathematical language. 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. Call students' attention to any words or phrases that helped to clarify the original statement. This will provide more students with an opportunity to produce language as they interpret the reasoning of others.
Design Principle(s): Support sense-making


aligns: yes