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: 
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: 
Construct a tangent line from a point outside a given circle to the circle.

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: 
No. This task does not involve drawing lines that are tangent to circles. The only line that may be drawn is the ray between points P and B.

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_pos_0

HS_1

im_task_001160_neg_0

HS_2
standard_description: Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).
problem_activity: Optional activity
The purpose of this activity is to provide a context where a ratio of fractions arises naturally, and students need to find an equivalent ratio to solve the problem. The ratio
\(2 \frac12: 1 \frac34\)
is equivalent to
\(10:7\)
, so a scaled copy of the Mona Lisa that is 10 inches by 7 inches would fit on the cover of the notebook. Other answers are possible. Some students might try to find the biggest possible copy that will fit on the cover, which would result in a different scale factor.
The digital version of the student materials includes an applet so that students can experiment with the context, because there are many related measurements within the context that can be hard to visualize. For example, the applet makes it clear that you can't simply scale down the Mona Lisa and make it perfectly fit on the notebook, since the notebook and the Mona Lisa are not scaled copies of each other. It also serves to remind students that the length and width of the Mona Lisa have to be scaled by the same factor, or the image becomes distorted.
Students discuss how they found their solution with a partner and determine if the scale factors they came up with are reasonable. Students must think about whether it makes sense to try to scale the picture so that it fills as much of the page as possible, or whether it makes more sense to leave room for a title. As students discuss with their partner, identify pairs of students who have a good argument that a certain scale factor makes more sense to use than another. Ask these students to share during the whole class discussion.
Digital
Launch
If desired, show students an image of the Mona Lisa.
If using the print version, and appropriate technology is available, consider displaying the applet for everyone to see to help students better understand the situation
ggbm.at/j8B9vZKV
.
The purpose of the applet is for experimenting and understanding the situation. If using it, demonstrate how it works, and ask students to think about:
How to use the applet to create scale copies of the Mona Lisa (both dimensions have to be adjusted by the same factor.)
Is it possible to scale down the Mona Lisa so that it perfectly covers the notebook? (No, choices have to be made about what the final product will look like.)
Arrange students in groups of 2. Give 3–5 minutes of quiet work time to do the problem. Then, ask them to take turns sharing with their partner the method used to calculate scale factor and reasonableness of their answers.
If using the digital activity, still have students work in groups of 2 and have them work individually on the problem with the applet, before sharing their method(s) to calculate scale factor with their partner.
Conversing, Representing, Writing: MLR2 Collect and Display.
As students discuss the scale factor they used, listen for and collect mathematical language students use to describe the strategies they used to find scale factors. Throughout the remainder of the lesson, continue to update collected student language and remind students to borrow language from the display as needed. This will help students use mathematical language during paired and group discussions.
Design Principle(s): Optimize output (for explanation); Maximize meta-awareness
Student Facing
In real life, the Mona Lisa measures
\(2 \frac12\)
feet by
\(1 \frac34\)
feet. A company that makes office supplies wants to print a scaled copy of the Mona Lisa on the cover of a notebook that measures 11 inches by 9 inches.
The applet is here to help you experiment with the situation. (It won't solve the problems for you.) Use the sliders to scale the image and drag the red circle to place it on the book. Measure the side lengths with the Distance or Length tool.
What size should they use for the scaled copy of the Mona Lisa on the notebook cover?
What is the scale factor from the real painting to its copy on the notebook cover?
Discuss your thinking with your partner. Did you use the same scale factor? If not, is one more reasonable than the other?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Print
Launch
If desired, show students an image of the Mona Lisa.
If using the print version, and appropriate technology is available, consider displaying the applet for everyone to see to help students better understand the situation
https://ggbm.at/j8B9vZKV
.
The purpose of the applet is for experimenting and understanding the situation. If using it, demonstrate how it works, and ask students to think about:
How to use the applet to create scale copies of the Mona Lisa. (Both dimensions have to be adjusted by the same factor.)
Is it possible to scale down the Mona Lisa so that it perfectly covers the notebook? (No, choices have to be made about what the final product will look like.)
Arrange students in groups of 2. Give 3–5 minutes of quiet work time to do the problem. Then, ask them to take turns sharing with their partner the method used to calculate scale factor and reasonableness of their answers.
If using the digital activity, still have students work in groups of 2 and have them work individually on the problem with the applet, before sharing their method(s) to calculate scale factor with their partner.
Conversing, Representing, Writing: MLR2 Collect and Display.
As students discuss the scale factor they used, listen for and collect mathematical language students use to describe the strategies they used to find scale factors. Throughout the remainder of the lesson, continue to update collected student language and remind students to borrow language from the display as needed. This will help students use mathematical language during paired and group discussions.
Design Principle(s): Optimize output (for explanation); Maximize meta-awareness
Student Facing
In real life, the Mona Lisa measures
\(2 \frac12\)
feet by
\(1 \frac34\)
feet. A company that makes office supplies wants to print a scaled copy of the Mona Lisa on the cover of a notebook that measures 11 inches by 9 inches.
What size should they use for the scaled copy of the Mona Lisa on the notebook cover?
What is the scale factor from the real painting to its copy on the notebook cover?
Discuss your thinking with your partner. Did you use the same scale factor? If not, is one more reasonable than the other?
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 might get stuck thinking the scaled copy needs to measure 11 inches by 9 inches.  Ask students:
Does the copy of the painting have to cover the entire notebook?
What are some other options if the image doesn't cover the entire notebook?
What if the image is bigger than the notebook cover? What if it is smaller?
Activity Synthesis
Select previously identified students to share the arguments they had with their partners.  Some guiding questions:
What scale factor did you and your partner agree upon? How did you both agree upon this?
Were there any similarities between the methods you and your partner used?  Were there any differences?


aligns: no