few_shots: Example:
Problem/activity:
Task
 Arrange the fractions in order from least to greatest. Explain your answer with a picture.
 $\frac{1}{5}, \frac{1}{7}, \frac{1}{3} $
 $\frac{2}{5}, \frac{2}{7}, \frac{2}{3} $
 $\frac{5}{6}, \frac{3}{6}, \frac{1}{6} $
 $\frac{5}{12}, \frac{8}{12}, \frac{4}{12} $

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. Apply and extend previous understandings of multiplication and division to multiply and divide fractions.
E. 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.
F. Develop Understanding of fractions as numbers.
Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8.


Your response, in json format:
{
  "explanation": "This problem involves understanding fractions as numbers by comparing their quantities.",
  "answer": "F"
}

Now, assign the following problem or activity to one or more relevant mathematical concepts/skills it teaches.
shot_info: im_task_000157

3
options: Represent and model with vector quantities.

Perform arithmetic operations with complex numbers.

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

Use complex numbers in polynomial identities and equations.

Reason quantitatively and use units to solve problems.

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

Extend the properties of exponents to rational exponents.

Perform operations on matrices and use matrices in applications.

Perform operations on vectors.

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

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

Represent complex numbers and their operations on the complex plane.

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

Use properties of rational and irrational numbers.
problem_activity: Activity
In this activity, students determine the length of a “diagonal” line segment on a grid. Students can give an exact value for the length of the line segment by finding the area of a square and writing the side length using square root notation. The goal of this activity is for students to connect values expressed using square roots with values expressed in decimal form—a form they are more familiar with.
Monitor for students who draw a tilted square for the first problem during the first two minutes of work time. Then monitor for students who use the following strategies to find the length of the segment:
drawing a square and finding the area
using tracing paper
using a compass to make a circle
Launch
For this activity, it is best if students do not have access to a calculator with a square root button. If student calculators do have a square root button that students are familiar with, tell students that their explanations about their answers to the second problem need to dig deeper than pressing a button. In later lessons, however, they will be able to use it.
Before students begin, remind students that “exact length” means it can’t just be an approximation, so if it is not a rational number, we should write it with square root notation. For example, a square with area 17 has a side length of exactly
\(\sqrt{17}\)
, which is a little larger than 4, since
\(4^2=16\)
.
Begin by displaying the diagram for all to see. Ask students how this diagram is similar and how it is different from the diagram in the warm-up. Then 2–3 minutes after students begin working, pause the class and select a previously identified student who drew a square on the grid to share what they did and why. Give 2–3 minutes work time to finish the problems followed by a whole-class discussion.
Representation: Internalize Comprehension.
Activate or supply background knowledge. Provide students with access to tracing paper, straight edges, and compasses to support information processing in estimating the length of the line segment.
Supports accessibility for: Visual-spatial processing; Organization
Student Facing
[IMG]
Estimate the length of the line segment to the nearest tenth of a unit (each grid square is 1 square unit).
Find the exact length of the segment.
Student Response
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Activity Synthesis
Select students to present in this sequence:
Someone who drew a square and used the area to find the exact side length.
Someone who used tracing paper. This is essentially like the number line as a ruler. Ask students what this tells us about the exact value we found with the square. (
\(\sqrt{10}\)
is about 3.1.)
Someone who used a compass to find the approximate side length. This is a more formal geometric construction, but it is just another way to use the number line as a ruler.
Speaking, Listening: MLR8 Discussion Supports.
As students share their methods for finding the exact length of the line segment, press for details in students’ reasoning by asking how they know the figure they drew in the coordinate plane is a square. Listen for and amplify the language students use to describe the important features of the square (e.g., opposite sides are equal, opposite sides are parallel, each angle is
\(90^\circ\)
). Then ask students to explain why the side length of the tilted square must be
\(\sqrt{10}\)
. This will support rich and inclusive discussion about strategies for finding the exact length of a line segment in the coordinate plane.
Design Principle(s): Support sense-making


correct_option_index: 9