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: 
Solve quadratic equations in one variable.

Answer: 
Yes. This task shows one quadratic equation with 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_pos_0

HS_1

im_task_001160_pos_0

HS_2
standard_description: Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.
problem_activity: Narrative
The purpose of this activity is for students to observe and use the structure of diagrams to find areas of shaded regions with non-unit fraction side lengths. Students build on what they learned in the previous activity, solidifying their understanding of why the numerator of a product of two fractions is the product of the numerators and the denominator of a product of fractions is the product of the denominators.
Action and Expression: Internalize Executive Functions.
Invite students to verbalize their strategy for writing multiplication expressions to represent the area of the shaded rectangle in each figure before they begin. Students can speak quietly to themselves, or share with a partner.
Supports accessibility for: Organization, Conceptual Processing, Language
Launch
Groups of 2
Activity
“Start working on completing the table independently. After a couple minutes, you'll work with your partner to complete the table and answer the rest of the questions.”
1–2 minutes: independent work time
5-8 minutes: partner work time
Monitor for students who:
notice the area of the shaded regions is always twentieths
write the expression
\(\frac{6}{5}\times\frac{4}{5}\)
to represent the shaded region of the last diagram in the table
explain that the expression
\(\frac{6\times4}{5\times4}\)
represents the shaded part of the last diagram in the table because
\(6\times4\)
represents the number of pieces that are shaded and
\(4 \times 5\)
represents the number of those pieces in the unit square
Student Facing
Complete the table.
{"table": [{"_attributes": {"border": "1", "class": ["c-table-spread"]}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col", "style": "width:45%"}, "_value": "diagram"}, {"_attributes": {"scope": "col", "style": "width:30%"}, "br": [{}], "_values": ["multiplication", "expression"]}, {"_attributes": {"scope": "col", "style": "width:25%"}, "br": [{}], "_values": ["shaded area", "(square units)"]}]}]}], "tbody": [{"tr": [{"td": [{"div": [{"_attributes": {"class": ["imgrid"]}, "div": [{"_attributes": {"class": ["g--row", "g--row-1"]}, "div": [{"_attributes": {"class": ["g--col-1", "g--column", "g--content", "g--full", "g--row-1"]}, "figure": [{"_attributes": {"class": ["embedded-content", "embedded-content-tikz-file", "embedded-content-image"], "data-embedder-target": "web", "data-record-id": "147492", "data-record-model": "TikzFile"}, "span": [{"_attributes": {"class": ["c-figure__label"]}, "_value": "A"}], "img": [{"_attributes": {"alt": "Square, length and width, 1. Partitioned into 4 rows of 5 of the same size rectangles. 6 rectangles shaded.", "role": "image", "src": "https://cms-im.s3.amazonaws.com/ui5g959sm5gv0dm8y5muaeuevuxb?response-content-disposition=attachment%3B%20filename%3D%22tikz-file-147492.svg%22%3B%20filename%2A%3DUTF-8%27%27tikz-file-147492.svg&response-content-type=image%2Fsvg%2Bxml&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240208%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240208T165553Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=da65b42438b700b09e62286557fa329bb057a4e3c43b71d5a338b3b21f3a4bad"}}], "figcaption": [{"_attributes": {"class": ["c-figcaption"]}}]}]}]}]}]}, {"_attributes": {"style": "text-align:justify"}}, {}]}, {"td": [{"div": [{"_attributes": {"class": ["imgrid"]}, "div": [{"_attributes": {"class": ["g--row", "g--row-1"]}, "div": [{"_attributes": {"class": ["g--col-1", "g--column", "g--content", "g--full", "g--row-1"]}, "figure": [{"_attributes": {"class": ["embedded-content", "embedded-content-tikz-file", "embedded-content-image"], "data-embedder-target": "web", "data-record-id": "149071", "data-record-model": "TikzFile"}, "span": [{"_attributes": {"class": ["c-figure__label"]}, "_value": "B"}], "img": [{"_attributes": {"alt": "Square, length and width, 1. Partitioned into 4 rows of 5 of the same size rectangles. 12 rectangles shaded.\u00a0", "role": "image", "src": "https://cms-im.s3.amazonaws.com/dx240g4b4kkym1gkgs7khx0yw7ri?response-content-disposition=attachment%3B%20filename%3D%22tikz-file-149071.svg%22%3B%20filename%2A%3DUTF-8%27%27tikz-file-149071.svg&response-content-type=image%2Fsvg%2Bxml&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240208%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240208T165553Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=216cb8ca62660faf0bab04b1fb04fd987604857004a334e4036d514267493e92"}}], "figcaption": [{"_attributes": {"class": ["c-figcaption"]}}]}]}]}]}]}, {}, {}]}]}]}]}
{"table": [{"_attributes": {"border": "1", "class": ["c-table-spread"]}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col", "style": "width:45%"}, "_value": "diagram"}, {"_attributes": {"scope": "col", "style": "width:30%"}, "br": [{}], "_values": ["multiplication", "expression"]}, {"_attributes": {"scope": "col", "style": "width:25%"}, "br": [{}], "_values": ["shaded area", "(square units)"]}]}]}], "tbody": [{"tr": [{"td": [{"div": [{"_attributes": {"class": ["imgrid"]}, "div": [{"_attributes": {"class": ["g--row", "g--row-1"]}, "div": [{"_attributes": {"class": ["g--col-1", "g--column", "g--content", "g--full", "g--row-1"]}, "figure": [{"_attributes": {"class": ["embedded-content", "embedded-content-tikz-file", "embedded-content-image"], "data-embedder-target": "web", "data-record-id": "149072", "data-record-model": "TikzFile"}, "span": [{"_attributes": {"class": ["c-figure__label"]}, "_value": "C"}], "img": [{"_attributes": {"alt": "Diagram. Square, length and width, 1. Partitioned into 4\u00a0rows of 5 of the same size rectangles. 20 rectangles shaded.", "role": "image", "src": "https://cms-im.s3.amazonaws.com/9mw9803l8i0vjwwj588r8xkqowrt?response-content-disposition=attachment%3B%20filename%3D%22tikz-file-149072.svg%22%3B%20filename%2A%3DUTF-8%27%27tikz-file-149072.svg&response-content-type=image%2Fsvg%2Bxml&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240208%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240208T165553Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=80f7eec3ac67d9991858cc362c8526df4c2ac740536b8a7383a5ba2034426a7e"}}], "figcaption": [{"_attributes": {"class": ["c-figcaption"]}}]}]}]}]}]}, {}, {}]}, {"td": [{"div": [{"_attributes": {"class": ["imgrid"]}, "div": [{"_attributes": {"class": ["g--row", "g--row-1"]}, "div": [{"_attributes": {"class": ["g--col-1", "g--column", "g--content", "g--full", "g--row-1"]}, "figure": [{"_attributes": {"class": ["embedded-content", "embedded-content-tikz-file", "embedded-content-image"], "data-embedder-target": "web", "data-record-id": "149073", "data-record-model": "TikzFile"}, "span": [{"_attributes": {"class": ["c-figure__label"]}, "_value": "D"}], "img": [{"_attributes": {"alt": "Diagram. Rectangle. Length, 2. Width, 1. Partitioned into 4 rows of 10 of the same size rectangles. 24 rectangles shaded.", "role": "image", "src": "https://cms-im.s3.amazonaws.com/wy0o9qune26s9u7htzi5ilsfmts4?response-content-disposition=attachment%3B%20filename%3D%22tikz-file-149073.svg%22%3B%20filename%2A%3DUTF-8%27%27tikz-file-149073.svg&response-content-type=image%2Fsvg%2Bxml&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240208%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240208T165553Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=57e693708d062f1a9de3113218a2c369f9dbe92ca8b9a751cd2729b2d6cbb65d"}}], "figcaption": [{"_attributes": {"class": ["c-figcaption"]}}]}]}]}]}]}, {}, {}]}]}]}]}
What patterns do you notice in the table?
Explain or show how the expression
\(\frac{6\times4}{5\times4}\)
represents the last diagram in the table.
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 explain that the product of the numerators represents the number of pieces in the shaded region and the product of the denominators represents the number of pieces in the whole, consider asking: “What is the same and what is different about diagrams A and B?”
Activity Synthesis
Ask previously selected students to share their reasoning.
“How do the expressions in the table represent the number of pieces shaded in and the size of the pieces shaded in?” (If we multiply the numerators, we get the number of pieces that are shaded in. If we multiply the denominators, we get the size of the pieces.)
Refer to diagrams and draw on each diagram to show how the multiplication of the numerators and denominators represents the number of shaded pieces and the size of the shaded pieces (that is, the number of those pieces in the whole).


aligns: yes