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: Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.
Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.

Understand a fraction a/b with a > 1 as a sum of fractions 1/b.
Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.
problem_activity: Narrative
This activity serves two main purposes. The first is to allow students to apply their understanding that the result of
\(a \times \frac{1}{b}\)
is
\(\frac{a}{b}\)
. The second is for students to reinforce the idea that any non-unit fraction can be viewed in terms of equal groups of a unit fraction and expressed as a product of a whole number and a unit fraction.
The activity uses a “carousel” structure in which students complete a rotation of steps. Each student writes a non-unit fraction for their group mates to represent in terms of equal groups, using a diagram, and as a multiplication expression. The author of each fraction then verifies that the representations by others indeed show the written fraction. A
s students discuss and justify their decisions they create viable arguments and critiqe one another’s reasoning (MP3).
MLR8 Discussion Supports.
Display sentence frames to support small-group discussion after checking their fraction diagram and equation: “I agree because . . .”, “I disagree because . . . .”
Advances: Conversing
Required Materials
Materials to Gather
Paper
Launch
Groups of 3
“Let's now use the patterns we saw earlier to write some true equations showing multiplication of a whole number and a fraction.”
Activity
3 minutes: independent work time on the first set of problems
2 minutes: group discussion
Select students to explain how they reasoned about the missing numbers in the equations.
If not mentioned in students' explanations, emphasize that: “We can interpret
\(\frac{5}{10}\)
as 5 groups of
\(\frac{1}{10}\)
,
\(\frac{8}{6}\)
as 8 groups of
\(\frac{1}{6}\)
, and so on.”
“In an earlier activity, we found that we can write any fraction as a multiplication of a whole number and a unit fraction. You'll now show that this is the case using fractions written by your group mates.”
Demonstrate the 4 steps of the carousel using
\(\frac{7}{4}\)
for the first step.
Read each step aloud and complete a practice round as a class.
“What questions do you have about the task before you begin?”
5–7 minutes: group work time
Student Facing
Use the patterns you observed earlier to complete each equation so that it’s true.
\(5 \times \frac{1}{10} =  \underline{\hspace{0.5in}}\)
\(8 \times \frac{1}{6} = \underline{\hspace{0.5in}}\)
\(4 \times  \underline{\hspace{0.5in}} = \frac{4}{5}\)
\(6 \times  \underline{\hspace{0.5in}} = \frac{6}{10}\)
\( \underline{\hspace{0.5in}} \times \frac{1}{4}= \frac{3}{4}\)
\( \underline{\hspace{0.5in}} \times \frac{1}{12}= \frac{7}{12}\)
Your teacher will give you a sheet of paper. Work with your group of 3 and complete these steps on the paper. After each step, pass your paper to your right.
Step 1: Write a fraction with a numerator other than 1 and a denominator no greater than 12.
Step 2: Write the fraction you received as a product of a whole number and a unit fraction.
Step 3: Draw a diagram to represent the expression you just received.
Step 4: Collect your original paper. If you think the work is correct, explain why the expression and the diagram both represent the fraction that you wrote. If not, discuss what revisions are needed.
Student Response
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Activity Synthesis
See lesson synthesis.


correct_option_index: 0