few_shots: 
Example 1:
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.

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


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:
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_neg_0

HS_0

im_task_000870_neg_0

HS_1

im_task_001160_pos_0

HS_2
standard_description: Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.
problem_activity: Narrative
Previously, students classified fractions based on their relationship to
\(\frac{1}{2}\)
and 1 (whether they are less than or more than these benchmarks). They used these classifications to compare fractions. In this activity, students are presented with fractions that are in the same group (for example, both less than
\(\frac{1}{2}\)
, or both greater than
\(\frac{1}{2}\)
but less than 1), so they need to reason in other ways to make comparisons.
Students can reason in a number of ways—by thinking about size and number of parts, drawing a diagram or number line, or reasoning numerically, but in most cases, they need to also rely on the idea of equivalence.
MLR7 Compare and Connect.
Synthesis: After each strategy has been presented, lead a whole-class discussion comparing, contrasting, and connecting the different approaches. Ask, “Did anyone solve the problem the same way, but would explain it differently?” and “Why did the different approaches lead to the same outcome?”
Advances: Representing, Conversing
Representation: Internalize Comprehension.
Activate background knowledge. Invite students to review the strategies they know for comparing fractions (reasoning about denominators or numerators, comparing to a benchmark, and writing equivalent fractions). Record students’ strategies on a visible display, including details (words or pictures) that will help them remember how to use the strategy.
Supports accessibility for: Conceptual Processing, Memory, Attention
Launch
Groups of 2
“Here are some fractions you’ve sorted in an earlier lesson. We compared them to
\( \frac{1}{2}\)
and 1.”
“What do the fractions in group 3 have in common? Why might they be in the same group?” (They are all greater than 1.)
“How are the fractions in group 1 different than those in group 2?” (Those in group 1 are less than
\( \frac{1}{2}\)
, and those in group 2 greater than
\( \frac{1}{2}\)
but less than 1.)
“We can tell that the fractions in group 2 are greater than those in group 1, and the fractions in group 3 are greater than those in the other groups.”
“Now compare the fractions in each group.”
Activity
“Take a few quiet minutes to work on the problems. Afterward, share your responses with your partner.”
7–8 minutes: independent work time
5 minutes: partner discussion
Monitor for students who:
reason by drawing number lines or tape diagrams
reason about the distance of each fraction from 0,
\(\frac{1}{2}\)
, or 1
reason about equivalent fractions (even if they don’t write out the multiplication numerically)
reason about equivalent fractions numerically by writing out the multiplication
Student Facing
Here are some pairs of fractions sorted into three groups. Circle the greater fraction in each pair. Explain or show your reasoning.
Group 1:
\(\frac{2}{10}\)
or
\(\frac{26}{100}\)
\(\frac{2}{5}\)
or
\(\frac{11}{100}\)
Group 2:
\(\frac{2}{3}\)
or
\(\frac{7}{12}\)
\(\frac{4}{5}\)
or
\(\frac{7}{10}\)
Group 3:
\(\frac{11}{5}\)
or
\(\frac{26}{10}\)
\(\frac{11}{3}\)
or
\(\frac{26}{12}\)
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Activity Synthesis
Select students who used different strategies to share their responses.
“How do we compare two fractions that are in the same group—say, both less than
\( \frac{1}{2}\)
or both greater than 1?” (We can think about how close or far away from
\( \frac{1}{2}\)
each fraction is.)
Highlight how equivalent fractions came into play in each strategy. For example, ask, “When comparing
\(\frac{2}{3}\)
and
\(\frac{7}{12}\)
, why was it helpful to think of the
\(\frac{2}{3}\)
as
\(\frac{8}{12}\)
?” Or, “When comparing
\(\frac{7}{10}\)
and
\(\frac{4}{5}\)
, why did you think of
\(\frac{4}{5}\)
as
\(\frac{8}{10}\)
?”
If no students mention that it is often easier to compare two fractions when they have the same denominator, ask them about it.


aligns: no