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

shot_info: fl_problem_001894_neg_0

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im_task_000870_neg_0

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im_task_001160_neg_0

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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 to give students opportunities to find the volumes of some cylinders. By finding the area of the base before finding the volume, students are encouraged to compute the volume by multiplying the area of its base by its height. This way of thinking about volume might be more intuitive for students than the formula
\(V=\pi r^2 h\)
. Notice students who plug the radius into a formula for the volume and students who find the area of the base and multiply that by the cylinder’s height.
The second problem of this activity focuses on exploring cylinders in a context. Generally, the volume of a container is the amount of space inside, but in this context, that also signifies the amount of material that fits into the space. Students are not asked to find the area of the base of the silo. Notice students that find the area before computing the volume and those that use the volume formula and solve for it directly. Notice at what step in the computations students approximate
\(\pi\)
.
When working with problems in a given context, it is sometimes convenient or practical to use an approximation of
\(\pi\)
. An example of this is given in the question regarding the volume of a grain silo and interpretations of the answer.
Launch
Provide access to colored pencils to shade the cylinder’s base. Give students 5–6 minutes of quiet work time followed by a whole-class discussion.
Representation: Internalize Comprehension.
Provide appropriate reading accommodations and supports to ensure students access to written directions, word problems and other text-based content.
Supports accessibility for: Language; Conceptual processing
Student Facing
Here is a cylinder with height 4 units and diameter 10 units.
[IMG]
Shade the cylinder’s base.
What is the area of the cylinder’s base? Express your answer in terms of
\(\pi\)
.
What is the volume of this cylinder? Express your answer in terms of
\(\pi\)
.
A silo is a cylindrical container that is used on farms to hold large amounts of goods, such as grain. On a particular farm, a silo has a height of 18 feet and diameter of 6 feet. Make a sketch of this silo and label its height and radius. How many cubic feet of grain can this silo hold? Use 3.14 as an approximation for
\(\pi\)
.
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?
One way to construct a cylinder is to take a rectangle (for example, a piece of paper), curl two opposite edges together, and glue them in place.
Which would give the cylinder with the greater volume: Gluing the two dashed edges together, or gluing the two solid edges together?
[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
Students might use the silo’s diameter instead of the radius to find the volume. Remind them of the formula for area of a circle and the discussion about diameter and radius earlier in this lesson.
Activity Synthesis
The goal of this discussion is to ensure students understand how to use the area of the cylinder’s base to calculate its volume. Consider asking the following questions:
“How does knowing the area of a circular base help determine the volume of a cylinder?” (The volume is this area multiplied by the height of the cylinder.)
“If the cylinder were on its side, how do you know which measurements to use for the volume?” (Since we need area of the base first, the radius or diameter of the circle will always be the measurement used for
\(r\)
, and the height is always the distance between the bases, or the measurement perpendicular to the bases. It doesn’t matter which direction the cylinder is turned.)
“When is it better to use approximations of pi instead of leaving it exact?” (Approximating pi helps us interpret an answer that has pi as a factor, like the area of a circular region or the volume of a cylindrical container.)
Speaking: MLR8 Discussion Supports.
Use this routine to support whole-class discussion. For each response that is shared, ask students to restate and/or revoice what they heard using mathematical language. Consider providing students time to restate what they hear to a partner, before selecting one or two students to share with the class. Ask the original speaker if their peer was accurately able to restate their thinking. Call students' attention to any words or phrases that helped to clarify the original statement.This will help students produce and make sense of the language needed to communicate about different strategies for calculating volume of cylinders.
Design Principle(s): Support sense-making; Optimize output (for explanation)


aligns: yes