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 division to divide unit fractions by whole numbers and whole numbers by unit fractions. (Students able to multiply fractions in general can develop strategies to divide fractions in general, by reasoning about the relationship between multiplication and division. But division of a fraction by a fraction is not a requirement at this grade.)

Interpret multiplication as scaling (resizing), by:
- Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n×a)/(n×b) to the effect of multiplying a/b by 1.
- Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication.

Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each person has a share of size 3/4. If 9 people want to share a 50-pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie?

Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.

Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.
problem_activity: Narrative
The goal of this activity is to use the distributive property to explain why multiplying a number by a fraction greater than one increases the size of the number while multiplying by a fraction less than one decreases the size of the number. Expressions are particularly useful here because they show explicitly how the size of the number relates to the product. For example writing
\(\frac{3}{5}\)
as
\(1 - \frac{2}{5}\)
and then multiplying by
\(\frac{4}{7}\)
gives:
\(\displaystyle \left(1 - \frac{2}{5}\right) \times \frac{4}{7} = \frac{4}{7} - \left(\frac{2}{5} \times \frac{4}{7}\right)\)
The revealing part of this calculation is that the structure of the right hand side shows that it is less than
\(\frac{4}{7}\)
without calculating the exact value (MP7). It must be less than
\(\frac{4}{7}\)
because it is
\(\frac{4}{7}\)
minus some other number.
Engagement: Internalize Self-Regulation.
Provide students an opportunity to self-assess and reflect on their own progress. For example, provide students with questions that relate to the size of the factors for them to reflect on once the activity is complete.
Supports accessibility for: Conceptual Processing, Attention, Memory
Launch
Groups of 2
Activity
1–2 minutes: quiet think time
8–10 minutes: partner work time
Monitor for students who use the expressions in the first problem to make the comparisons and then generalize about what happens when you multiply a number by any fraction greater than 1 or less than 1.
Student Facing
Rewrite each expression as a sum or difference of 2 products.
\(\left(1 - \frac{2}{5}\right) \times \frac{4}{7}\)
\(\left(1 + \frac{1}{5}\right)\times \frac{4}{7}\)
\(\left(1 - \frac{3}{8}\right)\times \frac{4}{7}\)
\(\left(1 + \frac{1}{8}\right)\times \frac{4}{7}\)
Fill in each blank with
\(<\)
or
\(>\)
to make the inequality true.
\(\left(1 - \frac{2}{5}\right)\times \frac{4}{7} \,\underline{\hspace{0.9cm}} \,\frac{4}{7}\)
\(\left(1 + \frac{1}{5}\right)\times \frac{4}{7} \,\underline{\hspace{0.9cm}}\, \frac{4}{7}\)
\(\left(1 - \frac{3}{8}\right)\times \frac{4}{7} \,\underline{\hspace{0.9cm}} \,\frac{4}{7}\)
\(\left(1 + \frac{1}{8}\right)\times \frac{4}{7} \,\underline{\hspace{0.9cm}} \,\frac{4}{7}\)
Describe the value of the product when
\(\frac{4}{7}\)
is multiplied by a fraction greater than 1. Explain your reasoning.
Describe the value of the product when
\(\frac{4}{7}\)
is multiplied by a fraction less than 1. Explain your reasoning.
Student Response
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Activity Synthesis
Invite students to share their expressions for the products in the first problem.
Display the equation:
\(\left(1 - \frac{2}{5}\right)\times \frac{4}{7} = \frac{4}{7} - \left(\frac{2}{5} \times \frac{4}{7}\right)\)
“How can you see that the value of the expression is less than
\(\frac{4}{7}\)
?” (It’s
\(\frac{4}{7}\)
minus something.)
“Does this reasoning also work for
\(\left(1 - \frac{3}{8}\right)\times \frac{4}{7}\)
?” (Yes, it’s again
\(\frac{4}{7}\)
minus some other number.)
“Will this reasoning work whenever you multiply a number less than 1 by
\(\frac{4}{7}\)
?” (Yes, I’ll always get
\(\frac{4}{7}\)
minus an amount so that’s less than
\(\frac{4}{7}\)
.)


correct_option_index: 1