few_shots: 
Example 1:
Problem/activity:
Find the general solution for:
$${\mathrm{cos}\left({\pi\over2}+x\right)=\mathrm{tan}{\pi\over4}}$$


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.

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


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).

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

Example 3:
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 $P$.


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.

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, does the problem or activity shown below enable students to learn the full intent of the following concept or skill?

shot_info: fl_problem_001894_neg_0

HS_0

im_task_000870_neg_0

HS_1

im_task_001160_pos_0

HS_2
standard_description: Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).
problem_activity: Narrative
In this activity, students continue to use the idea of partitioning a number line into smaller increments to reason about and generate equivalent fractions. Through repeated reasoning, students begin to see regularity in how the process of decomposing parts on a number line produces the numbers in the equivalent fractions (MP8). The task encourages students to think of the relationship between one denominator and the other in terms of factors or multiples (even if they don’t use those terms which connect to work in a previous unit).
Partitioning a number line into smaller parts becomes increasingly inconvenient when the denominator gets larger. As students begin to think about the relationship between tenths and hundredths, they see some practical limitations to using a number line to find equivalent fractions and are prompted to generalize the process of partitioning. (Students are not expected to draw a full number line with 100 parts.)
Launch
Groups of 2
Read the first problem as a class. Ask students to think quietly for a moment about whether what Priya wants to do can be done.
30 seconds: quiet think time
1 minute: partner discussion
Activity
“Take about 7–8 quiet minutes to work on the task. Afterwards, discuss your responses with your partner.”
7–8 minutes: independent work time
2–3 minutes: partner discussion
For the first problem, monitor for students who partition the lines by:
guessing and checking
reasoning multiplicatively (3 times what number gives 9, 10, or 12?) or in terms of multiples (Is 9, 10, or 12 a multiple of 3?)
reasoning in terms of division (9 divided by 3 is what number?) or in terms of factors (Is 3 a factor of 9, 10, or 12?)
For the second and third problem, monitor for students who find equivalent fractions for
\(\frac{1}{10}\)
by:
partitioning the number lines into smaller increments quantifying the new number of parts
finding multiples of 1 and 10, and using this strategy to write an equivalent fraction with denominator 100
Student Facing
Priya wants to find fractions that are equivalent to
\(\frac{2}{3}\)
, other than
\(\frac{4}{6}\)
. She wonders if she can find equivalent fractions with denominator 9, 10, and 12.
\(\frac{\phantom{000}}{9} \qquad \frac{\phantom{000}}{10} \qquad \frac{\phantom{000}}{12}\)
Can it be done? Use number lines to show your reasoning.
[IMG]
[IMG]
[IMG]
[IMG]
Represent
\(\frac{1}{10}\)
on a number line. Then, find two fractions that are equivalent to
\(\frac{1}{10}\)
. How would you use the number line to show that they are equivalent to
\(\frac{1}{10}\)
?
[IMG]
[IMG]
Can you find an equivalent fraction for
\(\frac{1}{10}\)
with 100 for the denominator? Explain or show your reasoning.
Student Response
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Advancing Student Thinking
If students conclude that the answer to the last question is “no, it can’t be done” because they find it impractical to partition each tenth on the number line into so many parts, ask them to visualize the line and think about how they’d partition it. Consider asking: “Into how many parts should each one-tenth section on the line be split to get hundredths? How do you know?”
Activity Synthesis
Invite previously identified students to share their strategies for answering the first two problems.
If not done by students in their explanations, consider asking students to revoice their reasoning in terms of factors and multiples.
See lesson synthesis.


aligns: no