few_shots: Example 1:
Problem/activity:
You’re mixing ingredients for cookies. The recipe says to combine 6 tablespoons, or ¾ stick, of butter with 1 cup of sugar. You accidentally mix in a whole stick of butter (8 tablespoons) with the cup of sugar. How can you fix this?


Mathematical concept/skill options: 
A. Analyze proportional relationships and use them to solve real-world and mathematical problems.
B. Understand ratio concepts and use ratio reasoning to solve problems.


Your response:
Thought: This problem focuses on the ratio between two ingredients. Students do not have to analyze proportional relationships in-depth, which is reserved for a higher grade level, but rather use ratios to find missing values.
Answer: B

Example 2:
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

Example 3:
Problem/activity:
Problem 1
The table below contains single equations in a single variable, single equations in two variables, and pairs of simultaneous equations in two variables. Check the appropriate box to show whether there is no solution, exactly 1 solution, or if there is an infinite number of solutions.
|                                                          | **           No solution          ** | **           Exactly 1 solution          ** | **           Infinite number of solutions          ** |
|----------------------------------------------------------|--------------------------------------|---------------------------------------------|-------------------------------------------------------|
|                       $${9=4x+7}$$                       |                                      |                                             |                                                       |
|                       $${y=3y+5}$$                       |                                      |                                             |                                                       |
|                    $${y={x\over2}}$$                     |                                      |                                             |                                                       |
|                     $${3x+6y-9=0}$$                      |                                      |                                             |                                                       |
|    $${x=y}$$                        $${1.25x=1.25y}$$    |                                      |                                             |                                                       |
| $${9.9=6x+8y}$$                        $${9x=2.5y-8.8}$$ |                                      |                                             |                                                       |
| $${11x-2y=1.5}$$                        $${11x-2y=2.5}$$ |                                      |                                             |                                                       |
Problem 2
Kim has a small container and a large container as shown.
[IMG]
It takes 16 of the small containers to fill the large container. Three small containers leave 1.95 gallons of space in the big container.
What is the size of each of the two containers?


Mathematical concept/skill options: 
A. Write expressions in equivalent forms to solve problems.
B. Understand properties of multiplication and the relationship between multiplication and division.
C. Interpret the structure of expressions.
D. Solve real-life and mathematical problems using numerical and algebraic expressions and equations.
E. Use polynomial identities to solve problems.
F. Understand the relationship between zeros and factors of polynomials.
G. Write and interpret numerical expressions.
H. Work with equal groups of objects to gain foundations for multiplication.
I. Perform arithmetic operations on polynomials.
J. Create equations that describe numbers or relationships
K. Generate and analyze patterns.
L. Analyze and solve linear equations and pairs of simultaneous linear equations.
M. Understand solving equations as a process of reasoning and explain the reasoning.
N. Apply and extend previous understandings of arithmetic to algebraic expressions.
O. Rewrite rational expressions.
P. Represent and solve problems involving multiplication and division.
Q. Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.
R. Reason about and solve one-variable equations and inequalities.
S. Understand and apply properties of operations and the relationship between addition and subtraction.
T. Understand the connections between proportional relationships, lines, and linear equations.
U. Multiply and divide within 100.
V. Gain familiarity with factors and multiples.
W. Use the four operations with whole numbers to solve problems.
X. Work with addition and subtraction equations.
Y. Represent and solve problems involving addition and subtraction.
Z. Solve problems involving the four operations, and identify and explain patterns in arithmetic.
AA. Solve equations and inequalities in one variable.
AB. Solve systems of equations.
AC. Represent and analyze quantitative relationships between dependent and independent variables.
AD. Work with radicals and integer exponents.
AE. Represent and solve equations and inequalities graphically.
AF. Analyze patterns and relationships.
AG. Use properties of operations to generate equivalent expressions.
AH. Add and subtract within 20.


Your response:
Thought: This set of problems involves solving linear equations and pairs of linear equations.
Answer: L

Now, assign the following problem or activity to one or more relevant mathematical concepts/skills it teaches.
shot_info: fl_problem_000972

6

fl_problem_001097

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fl_problem_001392

8
options: Apply trigonometry to general triangles.

Understand similarity in terms of similarity transformations.

Use coordinates to prove simple geometric theorems algebraically.

Apply geometric concepts in modeling situations.

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

Understand and apply the Pythagorean Theorem.

Find arc lengths and areas of sectors of circles.

Understand congruence in terms of rigid motions.

Prove geometric theorems.

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

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

Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.

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

Reason with shapes and their attributes.

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

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

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

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

Analyze, compare, create, and compose shapes.

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

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

Define trigonometric ratios and solve problems involving right triangles.

Experiment with transformations in the plane.

Make geometric constructions.

Explain volume formulas and use them to solve problems.

Understand and apply theorems about circles.

Prove theorems involving similarity.
problem_activity: Activity
The purpose of this activity is to let students determine how to use straightedge and compass moves to construct a regular hexagon precisely. Students should play with construction moves until they reach their goal rather than follow an explicit demonstration of construction steps. While the term
regular
appears in the task, it is not important for students to know the precise definition of
regular polygon
at this time.
Identify students whose explanations that the sides are congruent use tracing paper, or compare the radii of the different circles in the construction. Tracing paper connects to the idea of rigid motions, while comparing radii references the precise definition of a circle, which students will use throughout this unit and subsequent units.
Digital
Launch
Arrange students in groups of 2. Give students 10 minutes of work time followed by a whole-class discussion.
Engagement: Develop Effort and Persistence.
Encourage and support opportunities for peer interactions. Invite students to talk about their ideas with a partner before writing them down. Display sentence frames to support students when they explain their reasoning. For example, “Each of the sides of the shape I constructed are the same length because....”
Supports accessibility for: Language; Social-emotional skills; Conceptual processing
Student Facing
Here is a hexagon with all congruent angles and all congruent sides (called a
regular
hexagon).
Try to draw a copy of the regular hexagon using only the pen tool. Draw your copy next to the hexagon given, and then drag the given one onto yours. What happened?
Here is a figure that shows the first few steps to constructing the regular hexagon. Use straightedge and compass moves to finish constructing the regular hexagon. Drag the given one onto yours and confirm that it fits perfectly onto itself.
How do you know each of the sides of the shape are the same length? Show or 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.
Student Facing
Are you ready for more?
Why does the construction end up where it started? That is, how do we know the central angles go exactly 360 degrees around?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Print
Launch
Arrange students in groups of 2. Provide access to tracing paper. Give students 10 minutes of work time followed by a whole-class discussion.
Engagement: Develop Effort and Persistence.
Encourage and support opportunities for peer interactions. Invite students to talk about their ideas with a partner before writing them down. Display sentence frames to support students when they explain their reasoning. For example, “Each of the sides of the shape I constructed are the same length because....”
Supports accessibility for: Language; Social-emotional skills; Conceptual processing
Student Facing
Here is a hexagon with all congruent angles and all congruent sides (called a
regular
hexagon).
[IMG]
Draw a copy of the regular hexagon using only your pencil and no other tools. Trace your copy onto tracing paper. Try to fold it in half. What happened?
Here is a figure that shows the first few steps to constructing the regular hexagon. Use straightedge and compass moves to finish constructing the regular hexagon. Trace it onto tracing paper and confirm that when you fold it in half, the edges line up.
[IMG]
How do you know each of the sides of the shape are the same length? Show or 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.
Student Facing
Are you ready for more?
Why does the construction end up where it started? That is, how do we know the central angles go exactly 360 degrees around?
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
If students spend more than a few minutes without significant progress, tell them the segment given in the figure is one of the 6 sides of the hexagon. Invite students to compare the given hexagon to the start of the construction. Then ask if they can draw another segment to make an adjacent side of the hexagon.
Activity Synthesis
The purpose of this discussion is to build toward the concept of a proof by asking students to informally explain why a fact about a geometric object must be true. Ask previously identified students to share their responses to “How do you know each of the sides of the shape are the same length?”
Writing, Speaking, Conversing: MLR 1 Stronger and Clearer Each Time
. Use this with successive pair shares to give students a structured opportunity to revise and refine their response to “How do you know each of the sides of the shape are the same length?” Ask each student to meet with 2–3 other partners in a row for feedback. Provide students with prompts for feedback that will help individuals strengthen their ideas and clarify their language. For example, “Can you explain how…?”, “How do circles help with the construction?”, “What do you know about radii that helps here?”, or “What do you mean by…?”. Students can borrow ideas and language from each partner to strengthen their final explanation.
Design Principle(s): Optimize output (for justification); Support sense-making


correct_option_index: 22

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