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.


Your response:
{
  "explanation": "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.


Your response:
{
  "explanation": "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.


Your response:
{
  "explanation": "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: Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.

Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.
problem_activity: Activity
In this activity, students continue to develop their understanding of square roots. Students first find the areas of three squares, estimate the side lengths using tracing paper, and then write the exact side lengths. Then they make a table of side-area pairs. They then graph the ordered pairs from the table and use the graph to estimate the values of squares with non-integer side lengths, such as the ones they drew in the previous activity.
Digital
Launch
Arrange students in groups of 2. Provide access to calculators. Remind students about the meaning and use of square root notation. Have students work together on the first problem and check each other’s work. Have them make their graphs independently and then check with their partners that they look the same. Follow with a whole-class discussion.
Action and Expression: Internalize Executive Functions.
Chunk this task into more manageable parts. Check in with students after the first 2-3 minutes of work time. Check to make sure students have attended to all parts of the original figures.
Supports accessibility for: Organization; Attention
Student Facing
Find the area of each square and estimate the side lengths using your geometry toolkit. Then write the exact lengths for the sides of each square.
[IMG]
Complete the tables with the missing side lengths and areas.
{"table": [{"_attributes": {"border": "1"}, "tbody": [{"tr": [{"th": [{"_attributes": {"scope": "row"}}, {"_attributes": {"scope": "col"}, "_value": "side length,", "span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "833"}, "_value": "\\(s\\)"}]}], "td": [{"_value": "0.5"}, {}, {"_value": "1.5"}, {}, {"_value": "2.5"}, {}, {"_value": "3.5"}, {}]}, {"th": [{"_attributes": {"scope": "row"}, "_value": "row 1"}, {"_attributes": {"scope": "col"}, "_value": "area,", "span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "81"}, "_value": "\\(a\\)"}]}], "td": [{}, {"_value": "1"}, {}, {"_value": "4"}, {}, {"_value": "9"}, {}, {"_value": "16"}]}]}]}]}
{"table": [{"_attributes": {"border": "1"}, "tbody": [{"tr": [{"th": [{"_attributes": {"scope": "row"}}, {"_attributes": {"scope": "col"}, "_value": "side length,", "span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "833"}, "_value": "\\(s\\)"}]}], "td": [{"_value": "4.5"}, {}, {"_value": "5.5"}, {}, {"_value": "6.5"}, {}, {"_value": "7.5"}, {}]}, {"th": [{"_attributes": {"scope": "row"}, "_value": "row 1"}, {"_attributes": {"scope": "col"}, "_value": "area,", "span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "81"}, "_value": "\\(a\\)"}]}], "td": [{}, {"_value": "25"}, {}, {"_value": "36"}, {}, {"_value": "49"}, {}, {"_value": "64"}]}]}]}]}
Plot the points,
\((s, a)\)
, on the coordinate plane shown here.
Use this graph to estimate the side lengths of the squares in the first question. How do your estimates from the graph compare to the estimates you made initially using your geometry toolkit?
Use the graph to approximate
\(\sqrt{45}\)
.
Student Response
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click here to register or sign in
for free access to Student Response.
Print
Launch
Arrange students in groups of 2. Provide access to calculators. Remind students about the meaning and use of square root notation. Have students work together on the first problem and check each other’s work. Have them make their graphs independently and then check with their partners that they look the same. Follow with a whole-class discussion.
Action and Expression: Internalize Executive Functions.
Chunk this task into more manageable parts. Check in with students after the first 2-3 minutes of work time. Check to make sure students have attended to all parts of the original figures.
Supports accessibility for: Organization; Attention
Student Facing
Find the area of each square and estimate the side lengths using your geometry toolkit. Then write the exact lengths for the sides of each square.
[IMG]
Complete the tables with the missing side lengths and areas.
{"table": [{"_attributes": {"border": "1"}, "tbody": [{"tr": [{"th": [{"_attributes": {"scope": "col"}, "_value": "side length,", "span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "833"}, "_value": "\\(s\\)"}]}], "td": [{"_value": "0.5"}, {}, {"_value": "1.5"}, {}, {"_value": "2.5"}, {}, {"_value": "3.5"}, {}]}, {"th": [{"_attributes": {"scope": "col"}, "_value": "area,", "span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "81"}, "_value": "\\(a\\)"}]}], "td": [{}, {"_value": "1"}, {}, {"_value": "4"}, {}, {"_value": "9"}, {}, {"_value": "16"}]}]}]}]}
{"table": [{"_attributes": {"border": "1"}, "tbody": [{"tr": [{"th": [{"_attributes": {"scope": "col"}, "_value": "side length,", "span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "833"}, "_value": "\\(s\\)"}]}], "td": [{"_value": "4.5"}, {}, {"_value": "5.5"}, {}, {"_value": "6.5"}, {}, {"_value": "7.5"}, {}]}, {"th": [{"_attributes": {"scope": "col"}, "_value": "area,", "span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "81"}, "_value": "\\(a\\)"}]}], "td": [{}, {"_value": "25"}, {}, {"_value": "36"}, {}, {"_value": "49"}, {}, {"_value": "64"}]}]}]}]}
Plot the points,
\((s, a)\)
, on the coordinate plane shown here.
[IMG]
Use this graph to estimate the side lengths of the squares in the first question. How do your estimates from the graph compare to the estimates you made initially using your geometry toolkit?
Use the graph to approximate
\(\sqrt{45}\)
.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Activity Synthesis
Invite students to share their graphs from the second problem and display the graph for all to see. Ask: “What relationship does the graph display?” (The relationship between the side length and area of a square.) Select 2–3 students to share their answers to the last problem, including how they used the graph to estimate
\(s\)
for the tilted square they drew earlier.
Speaking: MLR8 Discussion Supports.
Use this routine to support whole-class discussion. After a student shares their response to the last problem, 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.
Design Principle(s): Support sense-making


correct_option_index: 0

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