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: 
Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).

Answer: 
No. This task does not involve an equation in two variables. There is only 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: 
Construct a tangent line from a point outside a given circle to the circle.

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, 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_neg_0

HS_1

im_task_001160_neg_0

HS_2
standard_description: Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x^(2) + y^(2))^(2) = (x^(2 )– y^(2))^(2) + (2xy)^(2 )can be used to generate Pythagorean triples.
problem_activity: Activity
In this activity, students calculate side lengths of similar triangles. They can use the scale factor between the similar triangles, studied in depth in previous lessons. Or they can look at internal ratios between corresponding side lengths within the triangles, introduced in the previous lesson. Students need to think strategically about which side lengths to calculate first since there are many missing values. As they discover more side lengths, this opens up more paths for finding the remaining values.
As students work, monitor for students who:
Use scale factors between triangles.
Notice that the long side is twice the short side in
\(GHI\)
and use that to find
\(c\)
,
\(d\)
, or
\(e\)
.
Notice that the long sides are equal in
\(ABC\)
and use that to find
\(h\)
,
\(d\)
, or
\(e\)
.
Select students who use different strategies to find side lengths to share during the discussion.
Launch
Tell students, “There are many ways to find the values of the unknown side lengths in similar triangles. Use what you have learned so far.” Give students 5 minutes of quiet work time followed by 5 minutes small group discussion and then a whole-class discussion.
Representation: Internalize Comprehension.
Demonstrate and encourage students to use color coding and annotations to highlight connections between representations in a problem. For example, use the same color for corresponding side lengths.
Supports accessibility for: Visual-spatial processing
Student Facing
Triangles
\(ABC\)
,
\(EFD\)
, and
\(GHI\)
are all similar. The side lengths of the triangles all have the same units. Find the unknown side lengths.
[IMG]
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
If students have trouble locating corresponding sides, suggest that they use tracing paper so they can rotate and or translate them. Another technique is to color corresponding side lengths the same color. For example, they could color
\(AB\)
,
\(EF\)
, and
\(GH\)
all red.
Activity Synthesis
Ask selected students to share the following strategies:
using (external) scale factors to move from one triangle to another
using quotients of corresponding side lengths within the triangles (internal scale factors)
Both methods are efficient and the method to use is guided by what information is missing and the numbers involved in the calculations. For example, if
\(h\)
is the first missing value we find, then comparing with triangle
\(ABC\)
and using internal scale factors is appropriate. To find
\(c\)
, again we can compare
\(ABC\)
and
\(GHI\)
and internal scale factors are appropriate again because
\(\frac{6}{5}\)
is half of
\(\frac{12}{5}\)
whereas comparing
\(\frac{12}{5}\)
and
\(4\)
is more involved (the scale factor is
\(\frac{5}{3}\)
from
\(\triangle GHI\)
to
\(\triangle ABC\)
).
Ask students to articulate how they knew which sides of the similar triangles correspond. Make sure to make the following reasoning pathways explicit for all:
Triangle
\(ABC\)
has two equal side lengths, so the other two triangles will as well. This insight is efficient for finding
\(h\)
.
One side of triangle
\(GHI\)
is twice the length of another side, so this will be true of the other triangles as well. This insight is helpful for finding
\(c\)
,
\(d\)
, and
\(e\)
.
Emphasize that there are many different relationships that can be used to find side lengths of similar triangles.
Speaking: MLR8 Discussion Supports:
To support students to articulate how they knew which sides of the similar triangles correspond, provide sentence frames such as, “For similar triangles _____ and _____ I know that sides _____ and _____ correspond because ________." Sentence frames invite and incentivize more student participation, conversation, and meta-awareness of language. This routine should help students reason about the ratio of corresponding sides of similar triangles and communicate their understanding.
Design Principle(s): Support sense-making, Optimize output (for comparison)


aligns: no