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 geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).
problem_activity: Activity
In the previous activity, students saw that subtracting the same value from all side lengths of a polygon did not produce a (smaller) scaled copy. This activity makes the case that adding the same value to all lengths also does not produce a (larger) scaled copy, reinforcing the idea that scaling involves multiplication.
This activity gives students a chance to draw a scaled copy without a grid and to use paper as a measuring tool. To create a copy using a scale factor of 2, students need to mark the length of each original segment and transfer it twice onto their drawing surface, reinforcing—in a tactile way—the meaning of scale factor. The angles in the polygon are right angles (and a 270 degree angle in one case) and can be made using the corner of an index card.
Some students may struggle to figure out how to use an index card or a sheet of paper to measure lengths. Before demonstrating, encourage them to think about how a length in the given polygon could be copied onto an index card and used as an increment for measuring. If needed, show how to mark the 4-unit length along the edge of a card and to use the mark to determine the needed lengths for the copy.
Teacher Notes for IM 6–8 Math Accelerated
This activity is optional. Use this activity if your students need more practice with the ideas from Which Operations? (Part 1).
Launch
Have students read the task statement and check that they understand which side of the polygon Andre would like to be 8 units long on his drawing. Provide access to index cards, so that students can use it as a measuring tool. Consider not explicitly directing students as to its use to give them a chance to use tools strategically (MP5). Give students 5–6 minutes of quiet work time, and then 2 minutes to share their work with a partner.
Student Facing
Andre wants to make a scaled copy of Jada's drawing so the side that corresponds to 4 units in Jada’s polygon is 8 units in his scaled copy.
Andre says “I wonder if I should add 4 units to the lengths of all of the segments?” What would you say in response to Andre? Explain or show your reasoning.
Create the scaled copy that Andre wants. If you get stuck, consider using the edge of an index card or paper to measure the lengths needed to draw the copy.
[IMG]
Student Response
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Student Facing
Are you ready for more?
The side lengths of Triangle B are all 5 more than the side lengths of Triangle A. Can Triangle B be a scaled copy of Triangle A? Explain your reasoning.
Student Response
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for free access to Extension Student Response.
Anticipated Misconceptions
Some students might not be convinced that making each segment 4 units longer will not work. To show that adding 4 units would work, they might simply redraw the polygon and write side lengths that are 4 units longer, regardless of whether the numbers match the actual lengths. Urge them to check the side lengths by measuring. Tell them (or show, if needed) how the 4-unit length in Jada’s drawing could be used as a measuring unit and added to all sides.
Other students might add 4 units to all sides and manage to make a polygon but changing the angles along the way. If students do so to make the case that the copy will not be scaled, consider sharing their illustrations with the class, as these can help to counter the idea that “scaling involves adding.” If, however, students do this to show that adding 4 units all around does work, address the misconception. Ask them to recall the size of corresponding angles in scaled copies, or remind them that angles in a scaled copy are the same size as their counterparts in the original figure.
Activity Synthesis
The purpose of the activity is to explicitly call out a potential misunderstanding of how scale factors work, emphasizing that scale factors work by multiplying existing side lengths by a common factor, rather than adding a common length to each.
Invite a couple of students to share their explanations or illustrations that adding 4 units to the length of each segment would not work (e.g. the copy is no longer a polygon, or the copy has angles that are different than in the original figure). Then, select a couple of other students to show their scaled copies and share how they created the copies. Consider asking:
What scale factor did you use to create your copy? Why?
How did you use an index card (or a sheet of paper) to measure the lengths for the copy?
How did you measure the angles for the copy?
Speaking: Math Language Routine 7 Compare and Connect.
This is the first time Math Language Routine 7 is suggested as a support in this course. In this routine, students are given a problem that can be approached using multiple strategies or representations, and are asked to prepare a visual display of their method. Students then engage in investigating the strategies (by means of a teacher-led gallery walk, partner exchange, group presentation, etc.), compare approaches, and identify correspondences between different representations. A typical discussion prompt is “What is the same and what is different?”, comparing their own strategy to the others. The purpose of this routine is to allow students to make sense of mathematical strategies by identifying, comparing, contrasting, and connecting other approaches to their own, and to develop students’ awareness of the language used through constructive conversations.
Design Principle(s): Maximize meta-awareness
How It Happens:
Use this routine to compare and contrast different methods for creating scaled copies of Jada’s drawing. Before selecting students to share a display of their method with the whole class, first give students an opportunity to do this in a group of 3–4.
Invite students to quietly investigate each other’s work. Ask students to consider what is the same and what is different about each display. Invite students to give a step-by-step explanation of their method using this sentence frame: “In order to create the copy, first I…. Next,…. Then, …. Finally,….”. Allow 1–2 minutes for each display and signal when it is time to switch.
Next, give each student the opportunity to add detail to their own display for 1-2 minutes. As students work on their displays, circulate the room to identify at least two different methods or two different ways of representing a method. Also look for methods that were only partially successful.
Consider selecting 1–2 students to share methods that were only partially successful in producing scaled copies. Then, select a couple of students to share displays of methods that did produce scaled copies.
Draw students’ attention to the approaches used in each drawing (e.g., adding the same value to each side length, not attending to the angles, multiplying by a common factor, not creating a polygon, etc.). Ask students, “Did this approach create a scaled copy? Why or why not?”
After the pre-selected students have finished sharing with the whole class, lead a discussion comparing, contrasting, and connecting the different approaches and representations.
In this discussion, demonstrate using the mathematical language “scale factor”, “corresponding”, and “multiplicative” to amplify student language.
Consider using these prompts:
“How did the scale factor show up in each method?”,
“Why did the different approaches lead to the same outcome?”,
“What worked well in _____’s approach/representation? What did not work well?”, and
“What role does multiplication play in each approach?”
Close the discussion by inviting 3 students to revoice the incorrect method for creating a scaled drawing, and then invite 3 different students to revoice the correct method for creating a scaled drawing. Then, transition back to the Lesson Synthesis and Cool Down.


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