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: 
Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.

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: 
Yes. This task asks students to draw a ray that bisects an angle. The act of drawing circles and rays may involve compass and straightedge or dynamic geometric software.

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_pos_0

HS_2
standard_description: Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison.
problem_activity: Narrative
In this activity, students see that they can find equivalent fractions by dividing the numerator and denominator by a common factor. They connect this strategy to the process of grouping unit fractions on a number line into larger equal-size parts. The result is a fewer number of parts, and smaller numbers for the numerator and denominator of equivalent fractions.
MLR8 Discussion Supports.
Synthesis: At the appropriate time, give students 2–3 minutes to make sure that everyone in their group can explain the strategies used in the given examples. Invite groups to rehearse what they will say when they share with the whole class.
Advances: Speaking, Conversing, Representing
Launch
Groups of 2
Activity
“Work with your partner to answer the first three problems.”
“Be prepared to explain how you think Andre’s and Kiran’s strategies are related.”
7–8 minutes: partner work time
“Take a few minutes to answer the last problem independently.”
2–3 minutes: independent work time for the last problem
Student Facing
Andre drew a number line and marked a point on it. Label the point with the fraction it represents.
[IMG]
To find other fractions that the point represents, Andre made copies of the number line. He drew darker marks on some of the existing tick marks.
Label the darker tick marks Andre made on each number line.
[IMG]
[IMG]
Kiran wrote the same fractions for the points but used a different strategy, as shown. Analyze his reasoning.
\(\frac{8 \ \div \ 4}{12 \ \div \ 4}=\frac{2}{3}\)
\(\frac{8 \ \div \ 2}{12 \ \div \ 2}=\frac{4}{6}\)
How do you think Andre’s and Kiran’s strategies are related?
Try using Kiran’s strategy to find one or more fractions that are equivalent to
\(\frac{10}{12}\)
and
\(\frac{18}{12}\)
.
Student Response
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Activity Synthesis
“What did Andre do with the number lines? How would it help him find equivalent fractions?” (Andre grouped the 12 parts into equal groups of different sizes—2s, 4s—to make bigger parts. Then, he counted the number of those new parts.)
“What did Kiran do? How is his strategy related to Andre’s?” (Kiran divided 12 by 4 and then by 2, similar to how Andre put 12 parts into groups of 4 and then of 2.)
“Notice that the equivalent fractions
\(\frac{2}{3}\)
and
\(\frac{4}{6}\)
both have smaller numbers for the numerator and denominator than the original fraction. Can you use Andre’s number line to show why this might be?” (The size of the parts are bigger, so there are fewer parts in 1 whole.)
If time permits, ask students:
“How many equivalent fractions can you find for
\(\frac{10}{12}\)
using Kiran’s way?” (One) “How many can you find for
\(\frac{18}{12}\)
?” (Three)
“What might be a reason that you could find more equivalent fractions for
\(\frac{18}{12}\)
than for
\(\frac{10}{12}\)
?” (18 and 12 have more factors in common than 10 and 12.)
“How would you show
\(\frac{9 \ \div \ 3}{12 \ \div \ 3}=\frac{3}{4}\)
on the number line?” (Put the original 12 parts into groups of 3 to get 4 parts, each being a fourth. Mark 3 of those 4 parts to show
\(\frac{3}{4}\)
.)


aligns: no