few_shots: Example:
Problem/activity:
For each pair of numbers, determine if the values are equivalent. Explain your reasoning for each one.
a.
$$-0.25$$
and
$$\frac{1}{-4}$$
b.
$$-\left ( \frac{1}{4} \right )$$
and
$$\frac{-1}{-4}$$
c.
$$\frac{-1}{4}$$
and
$$-\frac{1}{4}$$


Mathematical concept/skill options: 
A. Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.
B. Reason quantitatively and use units to solve problems.
C. Apply and extend previous understandings of numbers to the system of rational numbers.
D. Represent complex numbers and their operations on the complex plane.
E. Perform arithmetic operations with complex numbers.
F. Compute fluently with multi-digit numbers and find common factors and multiples.
G. Represent and model with vector quantities.
H. Perform operations on vectors.
I. Use properties of rational and irrational numbers.
J. Use complex numbers in polynomial identities and equations.
K. Extend the properties of exponents to rational exponents.
L. Know that there are numbers that are not rational, and approximate them by rational numbers.
M. Perform operations on matrices and use matrices in applications.
N. Apply and extend previous understandings of multiplication and division to divide fractions by fractions.


Your response:
Thought: This problem involves understanding fractions, multiplication, and division.
Answer: A

Now, assign the following problem or activity to one or more relevant mathematical concepts/skills it teaches.
shot_info: fl_problem_001097

7
options: Apply trigonometry to general triangles.

Experiment with transformations in the plane.

Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and spheres).

Analyze, compare, create, and compose shapes.

Find arc lengths and areas of sectors of circles.

Use coordinates to prove simple geometric theorems algebraically.

Reason with shapes and their attributes.

Visualize relationships between two-dimensional and three-dimensional objects.

Prove theorems involving similarity.

Understand and apply theorems about circles.

Graph points on the coordinate plane to solve real-world and mathematical problems.

Classify two-dimensional figures into categories based on their properties.

Apply geometric concepts in modeling situations.

Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.

Understand similarity in terms of similarity transformations.

Make geometric constructions.

Explain volume formulas and use them to solve problems.

Prove geometric theorems.

Draw and identify lines and angles, and classify shapes by properties of their lines and angles.

Translate between the geometric description and the equation for a conic section.

Draw, construct, and describe geometrical figures and describe the relationships between them.

Understand congruence and similarity using physical models, transparencies, or geometry software.

Define trigonometric ratios and solve problems involving right triangles.

Understand congruence in terms of rigid motions.

Solve real-world and mathematical problems involving area, surface area, and volume.

Understand and apply the Pythagorean Theorem.

Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.
problem_activity: Activity
In this activity, students continue to practice identifying corresponding parts of scaled copies. By organizing corresponding lengths in a table, students see that there is a single factor that relates each length in the original triangle to its corresponding length in a copy (MP8). They learn that this number is called a
scale factor
.
As students work on the first question, listen to how they reason about which triangles are scaled copies. Identify groups who use side lengths and angles as the basis for deciding. (Students are not expected to reason formally yet, but should begin to look to lengths and angles for clues.)
As students identify corresponding sides and their measures in the second and third questions, look out for confusion about corresponding parts. Notice how students decide which sides of the right triangles correspond.
If students still have access to tracing paper, monitor for students who use this tool strategically (MP5).
Launch
Arrange students into groups of 4. Assign each student one of the following pairs of triangles in the first question.
A and E
B and F
C and G
D and H
Give students 2 minutes of quiet think time to determine if their assigned triangles are scaled copies of the original triangle. Give another 2–3 minutes to discuss their responses and complete the first question in groups.
Discuss briefly as a class which triangles are scaled copies and select a couple of groups who reasoned in terms of lengths and angles to explain their reasoning. Some guiding questions:
What information did you use to tell scaled copies from those that are not?
How were you able to tell right away that some figures are not scaled copies?
Give students quiet work time to complete the rest of the task after the class recognizes that A, C, F, and H are not scaled copies.
Student Facing
Here is Triangle O, followed by a number of other triangles.
[IMG]
Your teacher will assign you two of the triangles to look at.
For each of your assigned triangles, is it a scaled copy of Triangle O? Be prepared to explain your reasoning.
As a group, identify
all
the scaled copies of Triangle O in the collection. Discuss your thinking. If you disagree, work to reach an agreement.
List all the triangles that are scaled copies in the table. Record the side lengths that correspond to the side lengths of Triangle O listed in each column.
| Triangle O |   |   |
|------------|---|---|
|            |   |   |
|            |   |   |
|            |   |   |
|            |   |   |
Explain or show how each copy has been scaled from the original (Triangle O).
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Student Facing
Are you ready for more?
Choose one of the triangles that is not a scaled copy of Triangle O. Describe how you could change at least one side to make a scaled copy, while leaving at least one side unchanged.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Anticipated Misconceptions
Students may think that Triangle F is a scaled copy because just like the 3-4-5 triangle, the sides are also three consecutive whole numbers. Point out that corresponding angles are not equal.
Activity Synthesis
Display the image of all triangles and invite a couple of students to share how they knew which sides of the triangles correspond. Then, display a completed table in the third question for all to see. Ask each group to present its observations about one triangle and how the triangle has been scaled from the original. Encourage the use of “corresponding” in their explanations. As students present, record or illustrate their reasoning on the table, e.g., by drawing arrows between rows and annotating with the operation students are describing, as shown here.
[IMG]
Use the language that students use to describe the side lengths and the numerical relationships in the table to guide students toward
scale factor
. For example: “You explained that the lengths in Triangle F are all twice those in the original triangle, so we can write those as “2 times” the original numbers. Lengths in Triangle A are half of those in the original; we can write “
\(\frac12\)
times” the original numbers. We call those multipliers—the 2 and the
\(\frac12\)
-scale factors. We say that scaling Triangle O by a scale factor of 2 produces Triangle F, and that scaling Triangle O by
\(\frac12\)
produces Triangle A.”


correct_option_index: 20