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: 
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_neg_0

HS_0

im_task_000870_pos_0

HS_1

im_task_001160_neg_0

HS_2
standard_description: Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.
problem_activity: Optional activity
In this Fermi problem, students estimate the total number of tiles needed to cover the Washington Monument.
Monitor for different approaches to solving the problem, and select students to share during the discussion. In particular, look for:
A range of estimates
Different levels of precision in reporting the final estimate
Different choices made to simplify the situation
Launch
Arrange students in groups of 1–4. Provide access to four-function calculators. If students are doing their own research, provide access to internet-enabled devices. If conducting a gallery walk at the end, provide access to tools for making a visual display.
Show a picture of the Washington Monument.
[IMG]
Before starting, ask students to come up with a guess how many tiles it would take to cover the Washington Monument and poll the class. Record all guesses.
Ask students to brainstorm the information they need to answer the question. Give the information provided when students ask for it, or provide access to internet-enabled devices so that students can find the information they need.
Vital information to have on hand includes:
Dimensions of the monument:
[IMG]
Standard sizes for square tiles: side lengths of 1 inch, 6 inches, 8 inches, 1 foot, and
\(1 \frac{1}{2}\)
feet.
Note that 55 feet is the height of the pyramid at the top, not the height of each triangular face. Students would need the Pythagorean Theorem to find the height of these faces, so either tell them it is close to 57.5 feet or they can use the 55 feet for their estimate. This does not have a significant impact on the area calculation, as the pyramid at the top does not account for most of the surface area.
Action and Expression: Internalize Executive Functions.
Provide students with a graphic organizer to capture information about the different assumptions they made when calculating the areas of the trapezoidal faces of the Washington Monument building.
Supports accessibility for: Language; Organization
Student Facing
How many tiles would it take to cover the Washington Monument?
[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
Students may not account for the faces of the monument that you cannot see in the picture.  Remind these students that some faces are hidden. If three-dimensional building blocks are available, consider having students who have difficulty visualizing the monument build a small model. It is difficult to model the exact shape of the Washington Monument, but the physical model may help students make the needed calculations.
Students may have difficulty converting units, especially if their tiles are measured in inches. Ask these students how many inches are in a foot and how many square inches are in a square foot.
Activity Synthesis
Invite students or groups to share different solution approaches. Consider asking students to create a visual display and conducting a gallery walk. Highlight different methods of calculating the area of the trapezoidal faces of the Washington Monument including:
Putting two trapezoid faces (or all 4 faces) together to make a parallelogram
Decomposing each trapezoid face into a rectangle and two triangles and then rearranging
Discuss why different people have different estimates. Some reasons to highlight include:
Using different-sized tiles to cover the Washington Monument
Thinking about how some tile is wasted, namely the tiles that go on the edge of each face
Rounding
This problem has a different flavor than the two preceding ones. The only place estimation comes into play is with how to cover the top triangles and perhaps thinking about what happens to tiles at the edge of a face. What all the problems share, however, is that we cannot give an
exact
answer. To provide a reasoned estimate, keeping track of all assumptions being made is vital. This tells us not only how accurately we should report our answer but also helps us communicate clearly to others who may have made different assumptions.
Speaking: MLR8 Discussion Supports.
Give students additional time to make sure that everyone in their group can explain their strategy for estimating the number of tiles using appropriate mathematical lanague (e.g., face, height, width, surface area, trapezoid, etc.). Be sure to vary who is called on to represent the ideas of each group. This routine will prepare students for the role of group representative and to support each other to take on that role.
Design Principle(s): Optimize output (for explanation)


aligns: yes