few_shots: 
Example:
Problem/activity:
Find the general solution for:
$${\mathrm{cos}\left({\pi\over2}+x\right)=\mathrm{tan}{\pi\over4}}$$


Concept/skill: 
Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

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, does the problem or activity shown below enable students to learn the full intent of the following concept or skill?

shot_info: fl_problem_001894_pos_0

HS_0
standard_description: Solve real-world and mathematical problems involving the four operations with rational numbers. (Computations with rational numbers extend the rules for manipulating fractions to complex fractions.)
problem_activity: Activity
The purpose of this activity is to see how to tell, from the equation, whether the sum will be positive, negative, or 0, without having to draw a number line diagram every time.
In this activity, students return to the context of height and depth to continue making sense of adding signed numbers. They use the structure of number line diagrams (MP7) to examine a variety of situations, which include starting in the positives, starting in the negatives, moving away from 0, moving towards 0, crossing over 0, and ending exactly on 0. From this variety of situations, students see that the lengths of the arrows in the number line diagrams give important information about the situation. This opens up the discussion of comparing the magnitude of the addends. When the addends have opposite signs, the longer arrow (the number with the larger magnitude) determines the sign of the sum.
\(400 + (\text-150) = 250\)
and
\(\text-200 + 150 = \text-50\)
.
This activity purposefully gives many problems to figure out, but only asks students to draw a diagram for 3 rows. Watch for students who think they need to draw a number line for every problem. Encourage them to look for and generalize from regularity in the problems (MP8) so that they don't need to always draw a diagram.
For students using the digital version of the materials, be sure they do not use the applet to represent and answer every question and bypass good thinking. Encourage students to figure out an answer for each question first, and then using the applet to check their work.
In preparation for dealing with subtraction of signed numbers, some of the problems involve determining how much the temperature changed, given the initial and final temperatures.
Digital
Launch
Explain that a
mountaineer
is someone who climbs mountains, and a
spelunker
is someone who explores caves and caverns. Arrange students in groups of 2. Give students 3 minute of quiet work time followed by partner discussion after the first question. Then have students work on the second question and follow with whole-class discussion.
Representation: Internalize Comprehension.
Represent the same information through different modalities by using physical objects to represent abstract concepts. Create an interactive display that allows students to experience physically moving an object (mountaineer) up and down a number line. Highlight connections between the act of physically moving the object up and down with arrows on a number line diagram.
Supports accessibility for: Conceptual processing; Visual-spatial processing
Student Facing
Explore the applet and then answer the questions.
A mountaineer is climbing on a cliff. She is 200 feet above the ground. If she climbs up, this will be a positive change. If she climbs down, this will be a negative change.
[IMG]
Complete the table.
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{}, {"_attributes": {"scope": "col"}, "br": [{}], "_values": ["starting elevation", "(feet)"]}, {"_attributes": {"scope": "col"}, "br": [{}], "_values": ["change", "(feet)"]}, {"_attributes": {"scope": "col"}, "br": [{}], "_values": ["final elevation", "(feet)"]}]}]}], "tbody": [{"tr": [{"th": [{"_value": "A"}], "td": [{"_value": "+200"}, {"_value": "75 up"}, {}]}, {"th": [{"_value": "B"}], "td": [{"_value": "+200"}, {"_value": "75 down"}, {}]}, {"th": [{"_value": "C"}], "td": [{"_value": "+200"}, {"_value": "200 down"}, {}]}, {"th": [{"_value": "D"}], "td": [{"_value": "+200"}, {}, {"_value": "+25"}]}]}]}]}
Write an addition equation and draw a number line diagram for B. Include the starting elevation, change, and final elevation in your diagram.
A spelunker is down in a cave next to the cliff. If she climbs down deeper into the cave, this will be a negative change. If she climbs up, whether inside the cave or out of the cave and up the cliff, this will be a positive change.
Complete the table.
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{}, {"_attributes": {"scope": "col"}, "br": [{}], "_values": ["starting elevation", "(feet)"]}, {"_attributes": {"scope": "col"}, "br": [{}], "_values": ["change", "(feet)"]}, {"_attributes": {"scope": "col"}, "br": [{}], "_values": ["final elevation", "(feet)"]}]}]}], "tbody": [{"tr": [{"th": [{"_value": "A"}], "td": [{"_value": "-20"}, {"_value": "15 down"}, {}]}, {"th": [{"_value": "B"}], "td": [{"_value": "-20"}, {"_value": "10 up"}, {}]}, {"th": [{"_value": "C"}], "td": [{"_value": "-20"}, {"_value": "20 up"}, {}]}, {"th": [{"_value": "D"}], "td": [{"_value": "-20"}, {"_value": "25 up"}, {}]}, {"th": [{"_value": "E"}], "td": [{"_value": "-20"}, {}, {"_value": "-50"}]}]}]}]}
Write an addition equation and draw a number line diagram for C and D. Include the starting elevation, change, and final elevation in your diagram.
​What does the expression
\(\text-45 + 60\)
tell us about the spelunker? What does the value of the expression tell us?
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
Explain that a
mountaineer
is someone who climbs mountains, and a
spelunker
is someone who explores caves and caverns. Arrange students in groups of 2. Give students 3 minute of quiet work time followed by partner discussion after the first question. Then have students work on the second question and follow with whole-class discussion.
Representation: Internalize Comprehension.
Represent the same information through different modalities by using physical objects to represent abstract concepts. Create an interactive display that allows students to experience physically moving an object (mountaineer) up and down a number line. Highlight connections between the act of physically moving the object up and down with arrows on a number line diagram.
Supports accessibility for: Conceptual processing; Visual-spatial processing
Student Facing
A mountaineer is climbing on a cliff. She is 400 feet above the ground. If she climbs up, this will be a positive change. If she climbs down, this will be a negative change.
Complete the table.
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{}, {"_attributes": {"scope": "col"}, "br": [{}, {}], "_values": ["starting", "elevation", "(feet)"]}, {"_attributes": {"scope": "col"}, "br": [{}], "_values": ["change", "(feet)"]}, {"_attributes": {"scope": "col"}, "br": [{}, {}], "_values": ["final", "elevation", "(feet)"]}]}]}], "tbody": [{"tr": [{"th": [{"_value": "A"}], "td": [{"_value": "+400"}, {"_value": "300 up"}, {}]}, {"th": [{"_value": "B"}], "td": [{"_value": "+400"}, {"_value": "150\u00a0down"}, {}]}, {"th": [{"_value": "C"}], "td": [{"_value": "+400"}, {"_value": "400\u00a0down"}, {}]}, {"th": [{"_value": "D"}], "td": [{"_value": "+400"}, {}, {"_value": "+50"}]}]}]}]}
[IMG]
Write an addition equation and draw a number line diagram for B. Include the starting elevation, change, and final elevation in your diagram.
[IMG]
​
A spelunker is down in a cave next to the cliff. If she climbs down deeper into the cave, this will be a negative change. If she climbs up, whether inside the cave or out of the cave and up the cliff, this will be a positive change.
Complete the table.
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{}, {"_attributes": {"scope": "col"}, "br": [{}], "_values": ["starting elevation", "(feet)"]}, {"_attributes": {"scope": "col"}, "br": [{}], "_values": ["change", "(feet)"]}, {"_attributes": {"scope": "col"}, "br": [{}], "_values": ["final elevation", "(feet)"]}]}]}], "tbody": [{"tr": [{"th": [{"_value": "A"}], "td": [{"_value": "-200"}, {"_value": "150 down"}, {}]}, {"th": [{"_value": "B"}], "td": [{"_value": "-200"}, {"_value": "100 up"}, {}]}, {"th": [{"_value": "C"}], "td": [{"_value": "-200"}, {"_value": "200 up"}, {}]}, {"th": [{"_value": "D"}], "td": [{"_value": "-200"}, {"_value": "250 up"}, {}]}, {"th": [{"_value": "E"}], "td": [{"_value": "-200"}, {}, {"_value": "-500"}]}]}]}]}
Write an addition equation and draw a number line diagram for C and D. Include the starting elevation, change, and final elevation in your diagram.
[IMG]
​
[IMG]
​
What does the expression
\(\text-75 + 100\)
tell us about the spelunker? What does the value of the expression tell us?
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 struggle with working backwards to fill in the change when given the final elevation. Ask them "What do you do to 400 to get to 50?" for example.
Activity Synthesis
The most important thing for students to get out of this activity is how to tell from the equation whether the sum will be positive, negative, or 0, without having to draw a number line diagram every time.
Write all of the equations where the two addends have the same sign. The ask, “What happens when the two addends have the same sign?”
Write all of the equations where the two addends have the opposite sign. The ask, “What happens when the two addends have opposite signs and . . .
the number with the larger magnitude is positive?”
the number with the larger magnitude is negative?”
Ask, “How can you tell when the sum will be zero?”
Speaking, Listening, Writing: MLR1 Stronger and Clearer Each Time.
To provide students with an opportunity to generalize about the sum of two addends, ask students to draft an initial response to the question “How can you tell from the equation whether the sum will be positive, negative, or 0?”. Ask students to meet with 2–3 partners for feedback. Provide students with prompts for feedback that will help each other strengthen their ideas and clarify their language (e.g., “What if the signs of both numbers are the same/different?” or “How do you know if the sum will be positive/negative?”, etc.). Students can borrow ideas and language from each partner to refine their explanation. This will help students to use mathematical language to generalize about the sums of rational numbers.
Design Principle(s): Optimize output (for generalization)


aligns: no