few_shots: Example:
Problem/activity:
Task
Cruz and Erica were both getting ready for soccer.
Cruz ran 1 lap around the school.
Erica ran 3 laps around the playground.
Erica said,
I ran more laps, so I ran farther.
Cruz said,
4 laps around the school is 1 mile, but it takes 12 laps around the playground to go 1 mile. My laps are much longer, so I ran farther.
Who is right? Draw a picture to help you explain your answer.


Mathematical concept/skill options: 
A. Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.
Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.
B. Extend understanding of fraction equivalence and ordering.
Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.
C. Use equivalent fractions as a strategy to add and subtract fractions.
D. Develop Understanding of fractions as numbers.
Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8.
E. Apply and extend previous understandings of multiplication and division to multiply and divide fractions.
F. Understand decimal notation for fractions, and compare decimal fractions.
Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.


Your response, in json format:
{
  "explanation": "In this problem, students learn to recognize equivalent fractions. That is, three laps around the playground is 3/12 mile, which is equivalent to 1/4, which is one lap around the school.",
  "answer": "B"
}

Now, assign the following problem or activity to one or more relevant mathematical concepts/skills it teaches.
shot_info: im_task_000212

4
options: Extend the properties of exponents to rational exponents.

Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.

Represent and model with vector quantities.

Apply and extend previous understandings of multiplication and division to divide fractions by fractions.

Perform arithmetic operations with complex numbers.

Know that there are numbers that are not rational, and approximate them by rational numbers.

Reason quantitatively and use units to solve problems.

Compute fluently with multi-digit numbers and find common factors and multiples.

Apply and extend previous understandings of numbers to the system of rational numbers.

Use properties of rational and irrational numbers.

Perform operations on matrices and use matrices in applications.

Perform operations on vectors.

Use complex numbers in polynomial identities and equations.

Represent complex numbers and their operations on the complex plane.
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)


correct_option_index: 1