few_shots: Example 1:
Problem/activity:
Find the general solution for:
$${\mathrm{cos}\left({\pi\over2}+x\right)=\mathrm{tan}{\pi\over4}}$$


Standard options: 
A. Prove the Pythagorean identity sin^(2)(θ) + cos^(2)(θ) = 1 and use it to find sin(θ), cos(θ),or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.
B. Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.


Thought: 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. Solving this problem does not require use of the Pythagorean identity.
Answer: B

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


Standard options: 
A. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
B. Solve quadratic equations in one variable.


Thought: This task shows one quadratic equation with one variable, x.
Answer: B

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 $P$.


Standard options: 
A. 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.
B. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.


Thought: 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.
Answer: A

Now, assign the following problem or activity to one or more relevant standards it aligns with.
shot_info: fl_problem_001894

HS_0

im_task_000870

HS_1

im_task_001160

HS_2
options: Solve real-world and mathematical problems involving the four operations with rational numbers. (Computations with rational numbers extend the rules for manipulating fractions to complex fractions.)

Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.

Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.
problem_activity: Activity
The purpose of this activity is to use the new skills of multiplying and dividing rational numbers to represent and solve problems in a new context (MP4). In this activity students are using using multiplication and division of negatives and working with a proportional relationship with a negative constant of proportionality. Students should use what they know about proportional relationships to help them (for example; points lie on a straight line, line passes through zero, there is a point
\((1, k)\)
that lies on the line).
Digital
Launch
Remind students that we can model positions below the surface with negative values, so drilling 30 feet down is represented with -30 feet.
Ask students what they remember about proportional relationships. Examples:
Often represented with an equation in the form
\(y=kx\)
.
The constant of proportionality, often called
\(k\)
, is the change in
\(y\)
for a change by 1 in
\(x\)
.
A graph representing a proportional relationship is a line through (0,0) and
\((1,k)\)
Arrange students in groups of 3 during the discussion.
Representation: Internalize Comprehension.
Provide appropriate reading accommodations and supports to ensure students have access to written directions, word problems and other text-based content.
Supports accessibility for: Language; Conceptual processing
Student Facing
A water well drilling rig has dug to a height of -60 feet after one full day of continuous use.
Assuming the rig drilled at a constant rate, what was the height of the drill after 15 hours?
If the rig has been running constantly and is currently at a height of -147.5 feet, for how long has the rig been running?
Use the coordinate grid to show the drill’s progress.
At this rate, how many hours will it take until the drill reaches -250 feet?
[IMG]
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Print
Launch
Remind students that we can model positions below the surface with negative values, so drilling 30 feet down is represented with -30 feet.
Ask students what they remember about proportional relationships. Examples:
Often represented with an equation in the form
\(y=kx\)
.
The constant of proportionality, often called
\(k\)
, is the change in
\(y\)
for a change by 1 in
\(x\)
.
A graph representing a proportional relationship is a line through (0,0) and
\((1,k)\)
Arrange students in groups of 3 during the discussion.
Representation: Internalize Comprehension.
Provide appropriate reading accommodations and supports to ensure students have access to written directions, word problems and other text-based content.
Supports accessibility for: Language; Conceptual processing
Student Facing
A water well drilling rig has dug to a height of -60 feet after one full day of continuous use.
Assuming the rig drilled at a constant rate, what was the height of the drill after 15 hours?
If the rig has been running constantly and is currently at a height of -147.5 feet, for how long has the rig been running?
Use the coordinate grid to show the drill’s progress.
[IMG]
At this rate, how many hours will it take until the drill reaches -250 feet?
[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
Some students may struggle to get the length of time out of the phrase "one full day of continuous use," wondering whether the drill was running at night. Let them know the drill has been going for 24 hours.
Activity Synthesis
Have students to share their solutions with each other in groups of three and work to come to agreement.
To wrap up, emphasize that because we can now complete any calculation with any rational number, we can extend constants of proportionality
\(k\)
to include negative values. Because the operations are now well-defined, we can answer questions using such a constant.
Speaking: MLR8 Discussion Supports.
Use this routine to support whole-class discussion. Invite select groups to share their solution and reasoning with the class. After each group presents, ask students to restate and/or revoice what they heard using mathematical language. Consider providing students time to restate what they heard to a partner, before selecting one or two students to share with the class. Ask the original group if this accurately reflects their thinking. This will provide more students with an opportunity to describe what they have learned about proportional relationships with a negative constant of proportionality.
Design Principle(s): Support sense-making


correct_option_index: 2