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 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_neg_0

HS_1

im_task_001160_neg_0

HS_2
standard_description: Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.
problem_activity: Warm-up
The purpose of this warm-up is to quickly remind students of different ways to write ratios. They also have an opportunity to multiply the number of each type of shape by 2 to make two copies of the flower, which previews the process introduced in this lesson for making a double batch of a recipe.
Launch
Arrange students in groups of 2. Ensure students understand there are 6 hexagons, 2 trapezoids, and 9 triangles in the picture, and that their job is to write ratios about the numbers of shapes. Give 2 minutes of quiet work time and then invite students to share their sentences with their partner, followed by whole-class discussion.
Student Facing
This flower is made up of yellow hexagons, red trapezoids, and green triangles.
[IMG]
Write sentences to describe the ratios of the shapes that make up this pattern.
How many of each shape would be in two copies of this flower pattern?
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 might get off track by attending to the area each shape covers. Clarify that this task is only concerned with the number of each shape and not the area covered.
Activity Synthesis
Invite a student to share a sentence that describe the ratios of shapes in the picture. Ask if any students described the same relationship a different way. For example, three ways to describe the same ratio are: The ratio of hexagons to trapezoids is
\(6:2\)
. The ratio of trapezoids to hexagons is 2 to 6. There are 3 hexagons for every trapezoid.
Ask a student to describe why two copies of the picture would have 12 hexagons, 4 trapezoids, and 18 triangles. If no student brings it up, be sure to point out that each number in one copy of the picture can be multiplied by 2 to find the number of each shape in two copies.


aligns: no