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: 
Solve quadratic equations in one variable.

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


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

HS_1

im_task_001160_pos_0

HS_2
standard_description: Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments.
problem_activity: Activity
This activity addresses the common misconception that if a figure is dilated by a factor of
\(k\)
, the image’s area also changes by a factor of
\(k\)
. As students decide which response (if either) they agree with, they are critiquing the reasoning of others (MP3).
Launch
Writing, Listening, Conversing: MLR1 Stronger and Clearer Each Time.
Use this routine to help students improve their written responses for the first question. Give students time to meet with 2–3 partners to share and receive feedback on their responses. Display feedback prompts that will help students strengthen their ideas and clarify their language. For example, “How is the area of the triangle affected when it is dilated by a factor of
\(k=0.25\)
?” or “How do you know that the area of the dilated triangle is 6.25 square inches?” Invite students to go back and revise or refine their written responses based on the feedback from peers. This will help students justify their reasoning for why the area of the triangle is multiplied by
\(k^2\)
when dilated by scale factor
\(k\)
.
Design Principle(s): Optimize output (for explanation); Cultivate conversation
Representation: Access for Perception.
Read the task statement aloud. Students who both listen to and read the information will benefit from extra processing time. Check for understanding by inviting students to rephrase the problem in their own words.
Supports accessibility for: Language
Student Facing
A triangle has area 100 square inches. It’s dilated by a factor of
\(k=0.25\)
.
Mai says, “The dilated triangle’s area is 25 square inches.”
Lin says, “The dilated triangle’s area is 6.25 square inches.”
For each student, decide whether you agree with their statement. If you agree, explain why. If you disagree, explain what the student may have done to arrive at their answer.
Calculate the area of the image if the original triangle is dilated by each of these scale factors:
\(k=9\)
\(k=\frac34\)
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Anticipated Misconceptions
Students may believe they can’t calculate the area of the dilated triangle if they don’t have its dimensions. Remind them of the two approaches highlighted in the warm-up and ask if either of those applies here.
Activity Synthesis
Here are some questions for discussion:
“Why didn’t we need the dimensions of the original triangle to calculate the area of the dilated triangle?” (We could multiply
\(k^2\)
by the original triangle’s area.)
“What are some different ways to deal with the scale factor
\(\frac34\)
?” (We can square the fraction to get
\(\left(\frac34 \right)^2 = \frac{9}{16}\)
, then multiply that fraction by the area. Or, we can divide 3 by 4 to get 0.75, square that to get 0.5625, then multiply that by 100.)


aligns: yes