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: Understand a fraction a/b with a > 1 as a sum of fractions 1/b.
Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.
problem_activity: Narrative
In this activity, students create a line plot using measurements to the nearest
\(\frac{1}{4}\)
and
\(\frac{1}{8}\)
inch. This task prompts students to use their understanding of fraction equivalence to plot and partition the horizontal axis.
Representation: Access for Perception.
Provide access to fraction strips that show fourths and eighths, and invite students to use them to answer question 4. Ask students to identify correspondences between the fraction strips and the horizontal axis of the line plot.
Supports accessibility for: Conceptual Processing, Visual-Spatial Processing
Launch
Groups of 2
“The table lists many different lengths.”
“What do you notice about the pencil lengths?” (Sample responses:
Some repeat more than one time.
The numbers are mixed numbers.
There are no whole numbers.)
1 minute: quiet think time
“There are some lengths that are more common or occur more often than others.”
“Tell a partner the length that is most common.” (
\(6\frac{3}{4}\)
)
Activity
Groups of 2
5 minutes: independent work time
Monitor for students who use equivalence to plot measurements to the nearest eighth inch.
“Share your line plots with your partner and make revisions as needed.”
2 minutes: partner discussion
Student Facing
Andre’s class measured the length of some colored pencils to the nearest
\(\frac{1}{4}\)
inch. The data are shown here:
\(1\frac{3}{4}\)
\(2\frac{1}{4}\)
\(5\frac{1}{4}\)
\(5\frac{1}{4}\)
\(4\frac{2}{4}\)
\(4\frac{2}{4}\)
\(6\frac{1}{4}\)
\(6\frac{3}{4}\)
\(6\frac{3}{4}\)
\(6\frac{3}{4}\)
[IMG]
Plot the colored-pencil data on the line plot.
[IMG]
Which colored-pencil length is the most common in the data set?
Write 2 new questions that could be answered using the line plot data.
Next, Andre’s class measured their colored pencils to the nearest
\(\frac{1}{8}\)
inch. The data are shown here:
\(1\frac{6}{8}\)
\(2\frac{2}{8}\)
\(5\frac{2}{8}\)
\(5\frac{3}{8}\)
\(4\frac{4}{8}\)
\(4\frac{4}{8}\)
\(6\frac{6}{8}\)
\(6\frac{6}{8}\)
\(6\frac{6}{8}\)
\(6\frac{3}{8}\)
Plot the colored-pencil data on the line plot.
[IMG]
Which colored-pencil length is the most common in the line plot?
Why did some colored-pencil lengths change on this line plot?
What is the difference between the length of the longest colored pencil and the shortest colored pencil? Show your reasoning.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Advancing Student Thinking
If students do not yet identify “most common” length as the length with the most data points on the line plot, consider asking: “Which measurement occurred more often than others?” and “Which pencil measurement was plotted most often?”
Activity Synthesis
“How many colored pencils were measured in Andre’s class?” (10, because there were ten data points and each represented a pencil that was measured.)
“What was the most common measurement in the first set of data? In the second set of data?” (In the first set of data:
\(6\frac{3}{4}\)
inches. In the second set of data:
\(6\frac{6}{8}\)
inches.)
“How did you use equivalence to help as you plotted measurement data in eighths of an inch?” (I know that two eighths are equivalent to one fourth and this helped to find eighths on the line plots.)
Use the line plot image to clearly label
\(\frac{1}{2}\)
,
\(\frac{1}{4}\)
and
\(\frac{1}{8}\)
with help from students.


aligns: yes