few_shots: 
Example 1:
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.

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


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:
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).

Problem/activity:
Task
Solve the quadratic equation
$$ x^2 = (2x - 9)^2 $$
using as many different methods as possible.


Answer: 
No. This task does not involve an equation in two variables. There is only 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_neg_0

HS_0

im_task_000870_neg_0

HS_1

im_task_001160_neg_0

HS_2
standard_description: Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.
problem_activity: Activity
The purpose of this activity is for students to calculate the volume of cones given their height and radius. Students are given a cylinder with the same height and radius and use the volume relationship they learned in the previous activity to calculate the volume of the cone. They then calculate the volume of a cone given a height and radius using the newly learned formula for volume of a cone. For the last problem, an image is not provided to give students the opportunity to sketch one if they need it.
Launch
Give students 3–5 minutes of quiet work time followed by a whole-class discussion.
Student Facing
Here is a cylinder and cone that have the same height and the same base area.
What is the volume of each figure? Express your answers in terms of
\(\pi\)
.
[IMG]
Here is a cone.
What is the area of the base? Express your answer in terms of
\(\pi\)
.
What is the volume of the cone? Express your answer in terms of
\(\pi\)
.
[IMG]
A cone-shaped popcorn cup has a radius of 5 centimeters and a height of 9 centimeters. How many cubic centimeters of popcorn can the cup hold? Use 3.14 as an approximation for
\(\pi\)
, and give a numerical answer.
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?
A grain silo has a cone shaped spout on the bottom in order to regulate the flow of grain out of the silo. The diameter of the silo is 8 feet. The height of the cylindrical part of the silo above the cone spout is 12 feet while the height of the entire silo is 16 feet.
How many cubic feet of grain are held in the cone spout of the silo? How many cubic feet of grain can the entire silo hold?
[IMG]
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
In the first problem, students might use 10 as the radius length. Ask students what the length of 10 in the picture is called. Ask students to recall the formula for the volume of a cylinder.
For students who are not sure where to begin the last problem since it does not have an image, encourage them to sketch and label their own.
Activity Synthesis
For the first problem, Invite students to explain how they calculated the volume of both figures and have them share the different strategies they used. If not mentioned by students bring up these strategies:
Calculate the volume of the cylinder, then divide volume of cylinder by 3 to get the volume of the cone.
Calculate the volume of the cylinder, then multiply volume of cylinder by
\(\frac13\)
to get the volume of the cone.
Calculate the volume of the cone, then multiply volume of cone by 3 to get the volume of the cylinder.
For the third problem, ask students to share any sketches they came up with to help them calculate the answer. Explain to students that sometimes we encounter problems that don’t have a visual example and only a written description. By using sketches to help to visualize what is being described in a problem, we can better understand what is being asked.
Engagement: Develop Effort and Persistence.
Break the class into small discussion groups and then invite a representative from each group to report back to the whole class. All group members should be prepared to share if invited.
Supports accessibility for: Language; Social-emotional skills; Attention
Writing: MLR3 Clarify, Critique, Correct.
Present an incorrect response to the first question that reflects a possible misunderstanding from the class. For example, “The volume is
\(4000\pi\)
because
\(\pi*4*10^2\)
.” Prompt students to critique the reasoning (e.g., ask, “Do you agree with the author’s reasoning? Why or why not?”) and then write feedback to the author that identifies the misconception and how to improve on his/her work. Listen for students who tie their feedback to the difference between the radius and diameter and use the academic vocabulary (e.g., height, radius, diameter, cylinder, cone, volume, etc.). This will help students evaluate, and improve on, the written mathematical arguments of others and highlight the distinction between and the importance of radii when calculating volume of cylinders and cones.
Design Principle(s): Maximize meta-awareness


aligns: yes