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: 
Solve quadratic equations in one variable.

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


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

Example 3:
Concept/skill: 
Construct a tangent line from a point outside a given circle to the circle.

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: 
No. This task does not involve drawing lines that are tangent to circles. The only line that may be drawn is the ray between points P and B.

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_pos_0

HS_1

im_task_001160_neg_0

HS_2
standard_description: Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., p^(2)). For example, by truncating the decimal expansion of Ö2, show that Ö2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.
problem_activity: Activity
In this activity students continue to build fluency operating with signed numbers as they match different expressions that have the same value. Students look for and use the relationship between inverse operations (MP7).
As students work, identify groups that make connections between the operations, for example they notice that subtracting a number is the same as adding its opposite, or that dividing is the same as multiplying by the multiplicative inverse.
Launch
Arrange students in groups of 2. Distribute pre-cut slips from the blackline master.
Student Facing
Your teacher will give you a set of cards. Group them into pairs of expressions that have the same value.
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
If students struggle to find matches, encourage them to think of the operations in different ways. You might ask:
How else can you think of ____ (subtraction, addition, multiplication, division)?
Activity Synthesis
Select students to share their strategies. Highlight stategies that compared the structure of the expressions instead of just the final answers.
Discuss:
Subtracting a number is equivalent to adding the additive inverse.
A number and its additive inverse have oppostie signs (but the same magnitude).
Dividing by a number is equivalent to multiplying by the multiplicative inverse.
A number and its multiplicative inverse have the same sign (but different magnitudes).
If desired, have students order their pairs of equivalent expressions from least to greatest to review ordering rational numbers.
Engagement: Develop Effort and Persistence.
Break the class into small discussion groups and then invite a representative from each group to report back to the whole class.
Supports accessibility for: Attention; Social-emotional skills
Speaking, Representing: MLR7 Compare and Connect.
Use this routine to support whole-class discussion. Ask students to consider what is the same and what is different about the structure of each expression. Draw students’ attention to the association between quantities (e.g., adding the additive inverse is subtracting; multiplying by the multiplicative inverse is dividing). These exchanges strengthen students’ mathematical language use and reasoning to describe when comparing inverse-operation relationships involving rational numbers.
Design Principle(s): Support sense-making; Maximize meta-awareness


aligns: no