few_shots: 
Example:
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.

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
standard_description: (+)Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant.
problem_activity: Activity
The purpose of this task is for students to understand that for a given number, numbers to the left are always less than the number, and numbers to the right are always greater than the number. The precise use of the term “absolute value” is not expected at this time.
Teacher Notes for IM 6–8 Accelerated
During the activity synthesis, introduce the terms
opposite
and
rational number
. Two numbers that are the same distance from zero on the number line are called opposites. Rational numbers are fractions and their opposites.
Digital
Launch
Allow 10 minutes quiet work time followed by whole-class discussion.
Students using the digital materials can graph the points and check them with the applet. Marks at each half, quarter, and eighth of a unit can be shown to help plot the points or to self-check for accuracy.
Representation: Internalize Comprehension.
Activate or supply background knowledge about equivalent fractions. Some students may benefit from access to pre-prepared number lines to compare fractions for the final two problems.
Supports accessibility for: Conceptual processing; Organization
Student Facing
Plot the numbers -2, 4, -7, and 10 on the number line. Label each point with its numeric value.
[IMG]
Decide whether each inequality statement is true or false. Be prepared to explain your reasoning.
\(\text-2 < 4\)
\(\text-2 < \text-7\)
\(4 > \text-7\)
\(\text-7 > 10\)
Drag each point to its proper place on the number line. Use your observations to help answer the questions that follow.
Andre says that
\(\frac14\)
is less than
\(\text{-}\frac {3}{4}\)
because, of the two numbers,
\(\frac14\)
is closer to 0. Do you agree? Explain your reasoning.
Answer each question. Be prepared to explain how you know.
Which number is greater:
\(\frac14\)
or
\(\frac54\)
?
Which number is farther from 0:
\(\frac14\)
or
\(\frac54\)
?
Which number is greater:
\(\text{-}\frac {3}{4}\)
or
\(\frac58\)
?
Which number is farther from 0:
\(\text{-}\frac {3}{4}\)
or
\(\frac58\)
?
Is the number that is farther from 0 always the greater number? Explain your reasoning.
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
Allow 10 minutes quiet work time followed by whole-class discussion.
Students using the digital materials can graph the points and check them with the applet. Marks at each half, quarter, and eighth of a unit can be shown to help plot the points or to self-check for accuracy.
Representation: Internalize Comprehension.
Activate or supply background knowledge about equivalent fractions. Some students may benefit from access to pre-prepared number lines to compare fractions for the final two problems.
Supports accessibility for: Conceptual processing; Organization
Student Facing
Plot the numbers -2, 4, -7, and 10 on the number line. Label each point with its numeric value.
[IMG]
Decide whether each inequality statement is true or false. Be prepared to explain your reasoning.
\(\text-2 < 4\)
\(\text-2 < \text-7\)
\(4 > \text-7\)
\(\text-7 > 10\)
Andre says that
\(\frac14\)
is less than
\(\text{-}\frac {3}{4}\)
because, of the two numbers,
\(\frac14\)
is closer to 0. Do you agree? Explain your reasoning.
Answer each question. Be prepared to explain how you know.
Which number is greater:
\(\frac14\)
or
\(\frac54\)
?
Which is farther from 0:
\(\frac14\)
or
\(\frac54\)
?
Which number is greater:
\(\text{-}\frac {3}{4}\)
or
\(\frac58\)
?
Which is farther from 0:
\(\text{-}\frac {3}{4}\)
or
\(\frac58\)
?
Is the number that is farther from 0 always the greater number? Explain your reasoning.
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 have difficulty comparing numbers on the negative side of the number line. Have students plot the numbers on a number line from least to greatest. It may be helpful to provide an example that students can use as a visual aid while they are working independently.
Some students may have difficulty comparing fractions in question 4. Remind them that comparing fractions is easier using a common denominator. Suggest that they subdivide the interval from 0 to 1 into fourths or eighths.
Activity Synthesis
The key takeaway from the discussion is that now that we have numbers on both sides of 0, distance from 0 isn’t enough to compare two numbers. Instead, we call numbers farther to the left on the number line “less” and numbers farther to the right “greater.” Display the following number line for all to see:
[IMG]
One inequality at a time, ask students to indicate whether they think each of the following is true or false:
\(\frac54 > \text{-}\frac {3}{2}\)
(true)
\(\frac54\)
is farther from 0 than
\(\text{-}\frac {3}{2}\)
(false)
\(\text{-}\frac {3}{2} < \text{-}\frac {3}{4}\)
(true)
\(\text{-}\frac {3}{2}\)
is farther from 0 than
\(\text{-}\frac {3}{4}\)
(true)
Invite students to share their reasoning. Here are some sentence frames that might be helpful:
“__ is greater (less) than __ because __.”
“__ is farther from 0 than __ because __.”
Speaking: MLR2 Collect and Display.
During the class discussion, record and display words and phrases that students use to explain why they decided certain inequality statements are true or false. Highlight phrases that include a reference to “to the right of,” “to the left of,” and a distance from zero. If students use gestures to support their reasoning, do your best to connect words to the gestures.
Design Principle(s): Optimize output (for explanation); Support sense-making


aligns: no