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: Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.
problem_activity: Optional activity
In both subsequent lessons and units, students will work extensively with similar right triangles. This activity gives students opportunities to practice the Pythagorean Theorem, and apply similar triangles in a contextual problem.
Launch
If students struggled to use the Pythagorean Theorem in a previous lesson:
Begin by asking students what they notice and wonder about the image from this activity.
Draw students’ attention to the right triangles, and ask them what must be true about the side lengths.
Provide some possible lengths for sides and have students practice finding an unknown side length in an isosceles right triangle to allow students to practice working with the square root of 2.
Speaking, Reading: MLR5 Co-Craft Questions.
Use this routine to increase awareness of the language used to talk about similar right triangles. Before revealing the questions in this activity, display the image of the quilt design made of right isosceles triangles. Ask students to write down possible mathematical questions that could be asked about the image. Invite students to compare their questions before revealing the actual questions. Listen for and amplify any questions about the features of the right triangles. For example, “Are all the right triangles similar?”, “What is the scale factor between the small and large right triangle?”, and “What are the side lengths of the right triangles?”
Design Principle(s): Maximize meta-awareness; Support sense-making
Student Facing
Here is a quilt design made of right isosceles triangles. The smallest squares in the center have an area of 1 square unit. Find the dimensions of the triangles.
[IMG]
Are the triangles similar? If so, what are the scale factors?
This quilt is meant for a baby (1 unit = 6 inches). To make a quilt for a queen-size bed, it needs to be 90 inches wide. What dimensions should the center squares of the big quilt have to reach that width?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Student Facing
Are you ready for more?
Here is a quilt design made of right triangles. Find as many different size triangles as you can.
[IMG]
Write similarity statements for 2 pairs of triangles.
The smallest triangles have legs 2 units and 3 units long. Write some equivalent ratios or equations that will help you determine the dimensions of triangle
\(LCQ\)
. Use your equivalent ratios or equations to find the dimensions of triangle
\(LCQ\)
.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Anticipated Misconceptions
Students who struggle with the number of similar triangles in the image can trace each size triangle separately onto tracing paper and label the corresponding sides using colored pencils.
Activity Synthesis
A key point to emphasize in this discussion is that ratios in which each ratio compares side lengths within each figure are also equivalent across similar figures. That is, for a triangle with side lengths
\(a, b, c\)
, and a similar triangle with corresponding side lengths
\(x,y,z\)
, we have equivalent ratios such as
\(\frac{a}{b} = \frac{x}{y}\)
.
Invite students to share strategies for scaling the figure up from a baby blanket to a quilt for a queen-size bed.


aligns: no