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: Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”).
problem_activity: Activity
In this activity, students practice using their understanding of ways to calculate the number of outcomes in the sample space without writing out the entire sample space (MP7). Many situations with multiple steps have very large sample spaces for which it is not helpful to write out the entire sample space, but it is still useful to know the number of outcomes in the sample space. In this activity, students find the number of different sandwiches that can be made from available options.
Launch
Explain to students that the sandwich makers are instructed to put a certain amount of each item on the sandwich for each selection. Therefore, if a person really loves tomatoes, he should ask for tomatoes twice as his two veggie choices. Give students 5 minutes of quiet work time followed by a whole-group discussion.
Action and Expression: Internalize Executive Functions.
Chunk this task into more manageable parts to support students who benefit from support with organizational skills in problem solving. For example, pause to check for understanding of determining the number of outcomes after 3–5 minutes of work time.
Supports accessibility for: Organization; Attention
Student Facing
A submarine sandwich shop makes sandwiches with one kind of bread, one protein, one choice of cheese, and
two
vegetables. How many different sandwiches are possible? Explain your reasoning. You do not need to write out the sample space.
Breads: Italian, white, wheat
Proteins: Tuna, ham, turkey, beans
Cheese: Provolone, Swiss, American, none
Vegetables: Lettuce, tomatoes, peppers, onions, pickles
[IMG]
Andre knows he wants a sandwich that has ham, lettuce, and tomatoes on it. He doesn’t care about the type of bread or cheese. How many of the different sandwiches would make Andre happy?
If a sandwich is made by randomly choosing each of the options, what is the probability it will be a sandwich that Andre would be happy with?
Student Response
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Student Facing
Are you ready for more?
Describe a situation that involves three parts and has a total of 24 outcomes in the sample space.
Student Response
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for free access to Extension Student Response.
Anticipated Misconceptions
Some students may attempt to write out the entire sample space. Encourage them to write a few outcomes to get the idea of what is possible, but let them know that the answer for question 1 is well over 1,000, so finding a pattern or way to calculate the answer might be more efficient.
Some students may not count the “none” option for cheese as a distinct choice. Show these students that a sandwich like “Italian bread, tuna, provolone, lettuce, and tomatoes” is different than a sandwich like “Italian bread, tuna, no cheese, lettuce, and tomatoes” and should be counted separately.
Some students may notice that the order of the vegetable selection may not matter. For example, selecting lettuce then tomato would create a similar sandwich to one with tomato then lettuce selected. Some sandwich shops may offer more of the first option, so we could ask students to consider the sandwiches as different based on this idea. Otherwise, there are only 720 sandwiches possible since there are only 15 different options for 2 vegetables and
\(3 \boldcdot 4 \boldcdot 4 \boldcdot 15 = 720\)
.
Activity Synthesis
The purpose of the discussion is to help students understand the calculations behind the solutions of these problems.
Some questions for discussion:
“Describe how the tree of sandwich options would look without drawing it out.” (The first column would have the 3 options for bread. Coming out from each of those options would be 4 branches for each of the proteins. From each of these there would be 4 more branches for each of the cheese options. Each of those would have 5 branches for the veggies. Finally, there would be 5 branches for the veggies again since the sandwich has 2 of them.)
“How is the tree connected to the calculation of the size of the sample space?” (Since the first two choices [bread and protein] are 3 groups of 4 branches, there are 12 options for those two choices. When adding the cheese option, there are 12 groups of 4 or 48 options. For the first veggie, there are 48 groups of 5 things or 240 options. Finally, there are 240 groups of 5 veggies for the last option, giving a total of 1,200 outcomes.)
“If the two veggie choices had to be different, would there be a higher or lower total number of possible sandwiches? Explain your reasoning.” (Fewer, since an item like “Italian bread, ham, Swiss, onions, and onions” was an option before, but is not an option with the new restriction.)
Speaking: MLR8 Discussion Supports.
Use this routine to support whole-class discussion. For each observation or response 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: no