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: 
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_pos_0

HS_0

im_task_000870_neg_0

HS_1

im_task_001160_neg_0

HS_2
standard_description: Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume.
problem_activity: Activity
In this activity, students get a chance to practice applying their skills in representing situations symbolically, and finding and interpreting solutions.
The structure of a row game supports students to check their thinking because their partners should get the same answer to different questions. When students’ answers disagree, they must make sense of each other’s questions and thinking to figure out whose answer is correct.
The practice prepares students to solve equations and solve for one variable in terms of another in their Algebra 1 lessons.
Monitor for students:
calculating some specific values before generating an equation
analyzing the quantities and relationships in the text
looking for and using mathematical structures, such as
\(ax + by\)
or
\(mx + b\)
Launch
Students will work independently to complete their designated column, and will work with their partners in the event that an answer is different from what their partners got.
Student Facing
Partner A completes only column A, and partner B completes only column B. Your answers for each problem should match. Work on one problem at a time, and check whether your answer matches your partner’s before moving on. If you don’t get the same answer, work together to find your mistake.
Column A:
Lin’s teacher has a daughter that is
\(\frac13\)
of his age. Write an expression to represent the daughter’s age. Let
\(z\)
represent the teacher’s age, in years.
Han wants to save
$
40. He hasn’t met his goal yet. Write an expression to represent how far Han is from his goal, in dollars. Let
\(q\)
represent the amount of money, in dollars, he’s saved so far.
Priya has some money to spend at a fair. It costs
$
6 to get in and
$
0.50 per ride ticket. Write an expression to represent how much Priya spends at the fair, in dollars. Let
\(x\)
represent the number of ride tickets Priya buys.
Diego is inviting some friends over to watch movies. He is buying popcorn and peanuts. Popcorn costs 6 cents per ounce and peanuts cost 17 cents per ounce. Write an expression to represent the total cost of peanuts and popcorn, in cents. Let
\(j\)
represent how many ounces of popcorn Diego buys and
\(k\)
represent how many ounces of peanuts he buys.
Column B:
Jada leaves the beach with some seashells. One out of every three of the shells turns out to contain a hermit crab. Write an expression to represent the number of hermit crabs Jada found. Let
\(z\)
represent the total number of seashells she collected.
Tyler started the school year with 40 pencils, but he’s lost some. Write an expression to represent how many pencils Tyler has left. Let
\(q\)
represent the number of pencils he’s lost so far.
When Clare bought her plant, it was 6 inches tall. Each week, it’s been growing
\(\frac12\)
of an inch. Write an expression to represent how tall Clare’s plant is, in inches. Let
\(x\)
represent the number of weeks since Clare bought her plant.
Mai is packing care packages. She is putting in boxes of granola bars that weigh 6 ounces each and paperback books that weigh 17 ounces each. Write an expression to represent the total weight of a care package, in ounces. Let
\(j\)
represent the number of boxes of granola bars and
\(k\)
represent the number of books.
Student Response
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Activity Synthesis
The goal of this discussion is for students to name and clarify strategies they have for writing algebraic expressions to describe given situations.
Strategies students might have include:
calculating some specific values before generating an expression
analyzing the quantities and relationships in the text
looking for and using mathematical structures, such as
\(ax + by\)
or
\(mx + b\)
Call on previously identified students to share their strategies for writing expressions, and support students to share insights or tips that will help other students. Possible questions for discussion include:
Why did you do calculations with some specific numbers first? How did that help you?
How did you record your calculations? What patterns did you notice?
When you read the situation, how many times did you read it?
What did you notice the first time you read the situation?
What did you notice when you looked at it again?
How did you decide what operations to use in your expression?
What made you think that your final expression would look like
\(mx + b\)
(or whatever structure is appropriate)?


aligns: no