few_shots: 
Example 1:
Problem/activity:
Find the general solution for:
$${\mathrm{cos}\left({\pi\over2}+x\right)=\mathrm{tan}{\pi\over4}}$$


Concept/skill: 
Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

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:
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 following problem or activity enable students to learn the full intent of the following concept or skill?

shot_info: fl_problem_001894_pos_0

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im_task_000870_neg_0

HS_1

im_task_001160_neg_0

HS_2
standard_description: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
problem_activity: Activity
In this activity students will apply their knowledge of trigonometry to a real-world context. They will use the safe ladder ratio to determine the safe ladder angle. Then they can use this angle to decide if a ladder is long enough for the given scenario.
Choose if students will use the ratio
\(4:1\)
or measure it themselves. Note that answers will vary if students measure. The image in the task is closer to
\(2:1\)
than
\(4:1\)
, so be prepared to discuss if that much variation is safe.
Launch
Tell students a ladder could tip over backwards if the angle is too large or slide down the wall if the angle is too small.
Provide students the safe ladder ratio by telling them: “A good rule of thumb for a safe angle to lean a ladder is the angle formed by your body when you stand on the ground and hold your arms out parallel to the ground. On most people that ratio is approximately
\(4:1\)
.”
Or if students will estimate the safe ladder ratio themselves, tell them: “A good rule of thumb for a safe angle to lean a ladder is the angle formed by your body when you stand on the ground and hold your arms out parallel to the ground. Form groups and measure the triangle one person’s body forms. That is, measure the distance from the floor to their shoulder. Then measure the distance from the front of their chest to their wrist.”
Speaking, Reading: MLR5 Co-Craft Questions.
Use this routine to help students interpret the language of ratios and angles in context. Display the image and the first sentence of this problem, without revealing the questions that follow. Ask students to write down possible mathematical questions that could be asked about the situation. Invite students to compare their questions with a partner before selecting 1–2 students to share their questions with the class. Listen for and amplify any questions that connect the ratios and angles to the context.
Design Principle(s): Maximize meta-awareness; Support sense-making
Engagement: Develop Effort and Persistence.
Encourage and support opportunities for peer collaboration. Prior to the whole-class discussion, invite students to share their work with a partner. Display sentence frames to support conversation such as: “First, I _____ because . . .”, “I noticed _____ so I . . .”, “Why did you . . .?”, “I agree/disagree because . . . .”
Supports accessibility for: Language; Social-emotional skills
Student Facing
A good rule of thumb for a safe angle to use when leaning a ladder is the angle formed by your body when you stand on the ground and hold your arms out parallel to the ground.
What are the angles in the triangle formed by your body and the ladder?
What are the angles in the triangle formed by the ladder, the ground, and the railing? Explain or show your reasoning.
You have a 13 foot long ladder and need to climb to a 12 foot tall roof.
If you put the top of the ladder at the top of the wall, what angle is formed between the ladder and the ground?
Is it possible to adjust the ladder to a safe angle? If so, give someone instructions to do so. If not, explain why not.
[IMG]
Student Response
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click here to register or sign in
for free access to Student Response.
Student Facing
Are you ready for more?
People have various proportions to their body. Suppose that someone’s height to arm ratio is
\(5:1\)
.
What are the angles in the triangle formed by their body and the ladder?
How far off is this from the
\(4:1\)
safe angle?
What could this person do to make the ladder closer to the safe ladder angle?
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
Some students might struggle to determine if the ladder can reach the roof. Inform them that the roof height has to be 12 feet but we can use as much of the ladder length as we want up to 13 feet. Invite them to draw a diagram where the ladder extends beyond the top of the building.
Activity Synthesis
Display the image for all to see and invite a student to demonstrate their explanation that the triangles are similar by annotating the image. Emphasize that all of trigonometry is based on similar right triangles. These are just two of the infinite set of right triangles with a
\(4:1\)
ratio and an angle measure of 76 degrees.


aligns: yes