few_shots: 
Example 1:
Concept/skill: 
Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

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


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:
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: 
Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.

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: 
Yes. This task asks students to draw a ray that bisects an angle. The act of drawing circles and rays may involve compass and straightedge or dynamic geometric software.

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_pos_0

HS_0

im_task_000870_neg_0

HS_1

im_task_001160_pos_0

HS_2
standard_description: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
problem_activity: Optional activity
In this activity, students are building skills that will help them in mathematical modeling (MP4). Asking students to request needed information invites them to identify their variables and helps students recall how much information is needed to determine a scale factor and apply it to a known distance to find the scaled distance. This activity also allows students to practice calculating proportional relationships in a scale drawing. Monitor for students who:
calculate using the equivalent ratios
\(\frac{\text{Earth’s scaled diameter}}{\text{Earth’s actual diameter}}=\frac{\text{planet’s scaled diameter}}{\text{planet’s actual diameter}}\)
calculate using the equivalent ratios
\(\frac{\text{Earth’s scaled diameter}}{\text{planet’s scaled diameter}}=\frac{\text{Earth’s actual diameter}}{\text{planet’s actual diameter}}\)
find a unit rate and scale up or down
calculate by finding a scale factor and multiplying their planet’s actual diameter by that scale factor
Making spreadsheet technology available gives students an opportunity to choose appropriate tools strategically (MP5).
Launch
Tell students they will be making a scale drawing of the solar system, with accurately scaled sizes and distances from Earth for each planet. Display a circle with 2 cm diameter and explain that it will represent Earth. Show students where the class will (attempt to) display the scale drawing of the solar system. Explain that each group will be assigned a planet to draw and add to the scale drawing, at the appropriate distance.
Arrange students in groups of 3. Assign each group a planet. If there aren't enough groups to do all the planets pick some planets that are smaller than Earth and some that are larger than Earth. It is also fine to have multiple groups work on the same planet.
Pause the class after students have determined what information they need. Ask groups what information they need. Display the table and tell students this should be enough to figure out all the information they asked for. Note that any reference to the distance from a planet to Earth assumes that all planets are aligned perfectly in a line in order of orbital distance to the Sun.
| planet  | average diameter (km) | average orbit radius (km) |
|---------|-----------------------|---------------------------|
| Mercury |                       |                           |
|  Venus  |                       |                           |
|  Earth  |                       |                           |
|  Mars   |                       |                           |
| Jupiter |                       |                           |
| Saturn  |                       |                           |
| Uranus  |                       |                           |
| Neptune |                       |                           |
Engagement: Develop Effort and Persistence.
Encourage and support opportunities for peer interactions. Prior to the whole-class discussion, invite students to share their work with a partner. Display sentence frames to support student conversation such as: “First, I _____ because . . .”, “I noticed_____ so I . . .”, “Why did you . . .?”
Supports accessibility for: Language; Social-emotional skills
Student Facing
The class is going to make a scale drawing of the planets in the solar system and their distances from Earth. Your teacher will assign you a planet to draw and place on the scale drawing. A circle with a diameter of 2 cm represents Earth.
What information do you need in order to draw your planet and place it accurately on the class’s scale drawing?
Your teacher will give you some information. Record the information here. Calculate any additional information you need.
Draw the scale drawing of your planet on a separate sheet of blank paper. Label it with measurements. When you’re finished, place it the correct distance from Earth on the class’s scale drawing.
You probably weren’t able to place your planet the correct scaled distance from Earth. Why not?
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?
What has the greater quotient: the radius of Earth compared to the Sun or the radius of the Sun compared to the radius of the star Betelgeuse?
If a circle of radius 6 cm represents the star Betelgeuse, how large would the Sun be? How large would the earth be?
Draw the scale drawing of the Sun and Earth, or explain why you can't.
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
If students struggle to use the information about orbital radius to find the planet’s distance from Earth encourage them to sketch the situation and label the information they know. (Image not to scale.)
[IMG]
Activity Synthesis
The key point of this discussion is to make sure students recall methods for finding unknown values in proportional relationships, and understand that the measurements in the scale drawing must all be scaled by the same scale factor.
Invite multiple students to share the calculations they did to find the size of their planet, given Earth’s scaled and actual size. First, select students to share who calculated using the equivalent ratios
\(\frac{\text{Earth’s scaled diameter}}{\text{Earth’s actual diameter}}=\frac{\text{planet’s scaled diameter}}{\text{planet’s actual diameter}}\)
or
\(\frac{\text{Earth’s scaled diameter}}{\text{planet’s scaled diameter}}=\frac{\text{Earth’s actual diameter}}{\text{planet’s actual diameter}}\)
. Next, select students to share who calculated by finding a scale factor or unit rate. Invite students to share how the scale factor is included in all the calculations. (The quotients in the equivalent ratios are the same as the scale factor.)
Ask students:
What units are your answers in? (Scaled values are in cm, actual values are in km.)
Why didn't you need to convert from kilometers to centimeters? (The scale factor includes the units already.)
How many times bigger is Earth than our 2 cm model? (A scale of
\(2 \text{ cm}:12,\!756 \text{ km}\)
is equivalent to a scale of
\(1 \text{ cm}:6,\!378 \text{ km}\)
which is equivalent to a scale of
\(1 \text{ cm}:637,\!800,\!000 \text{ cm}\)
. So it's 637,800,000 times bigger!)
Speaking: MLR8 Discussion Supports.
Use this routine to support whole-class discussion. As students share their strategies for calculating the size of their planet, ask students to restate what they heard using precise 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 that helped clarify the original explanation, such as “ratio,” “scale factor,” “scaled diameter,” and “actual diameter.” This provides more students with an opportunity to produce language as they interpret the reasoning of others.
Design Principle(s): Support sense-making


aligns: no