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: 
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_pos_0

HS_0

im_task_000870_pos_0

HS_1

im_task_001160_neg_0

HS_2
standard_description: Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.
problem_activity: Activity
In this activity, students get another opportunity to practice drawing dilations precisely. They also get to see the effects of different scale factors on an image. By assigning students scale factors less than and greater than one, students get a chance to see how the image is taken closer to or farther away from the center of dilation. By assigning equivalent scale factors, such as
\(\frac32\)
and 1.5, students get a chance to explore how images dilated with the same scale factor are congruent. They also get a chance to remind themselves of decimal representations of fractions, which will be useful when they calculate scale factors from ratios of side lengths in later lessons.
Launch
Arrange students in groups of 2. Assign each student a scale factor from the list:
\(\frac14\)
,
\(\frac12\)
,
\(\frac34\)
,
\(\frac76\)
,
\(\frac32\)
,
\(\frac52\)
, 0.25, 0.5, 1.5, 2.5
Representation: Internalize Comprehension.
Chunk this task into more manageable parts to differentiate the degree of difficulty or complexity. Assign
\(\frac12\)
and
\(\frac32\)
to a pair of students who would benefit from additional support.
Supports accessibility for: Conceptual processing; Organization; Attention
Student Facing
Here is a center of dilation and a triangle.
[IMG]
Measure the sides of triangle
\(EFG\)
(to the nearest mm).
Your teacher will assign you a scale factor. Predict the relative lengths of the original figure and the image after you dilate by your scale factor.
Dilate triangle
\(EFG\)
using center
\(C\)
and your scale factor.
How does your prediction compare to the image you drew?
Use tracing paper to copy point
\(C\)
, triangle
\(EFG\)
, and your dilation. Label your tracing paper with your scale factor.
Align your tracing paper with your partner’s. What do you notice?
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?
[IMG]
Dilate triangle
\(FEG\)
using center
\(C\)
and scale factors:
\(\frac12\)
2
What scale factors would cause some part of triangle
\(E'F'G'\)
’ to intersect some part of triangle
\(EFG\)
?
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
If students are unclear about relative lengths, tell them to make a prediction and write a description such as “a little smaller” or “a lot bigger” before they calculate.
Activity Synthesis
The goal of this discussion is to identify the effects of various scale factors. Collect and align several tracing papers on the original image. Invite students to make generalizations about different scenarios:
compare scale factors which are greater and less than one
compare equivalent scale factors with different representations (such as 0.5 and
\(\frac12\)
)
Speaking: MLR8 Discussion Supports.
Use this routine to help students produce statements about the scale factor of a dilation. Provide sentence frames for students to use when they generalize the effects of various scale factors, such as: “If _____ , then _____ because….” and “_____ will always _____ because….”
Design Principle(s): Support sense-making; Optimize output (for generalization)


aligns: no