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.


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.


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.


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

7

fl_problem_001392

8
options: Visualize relationships between two-dimensional and three-dimensional objects.

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

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

Use coordinates to prove simple geometric theorems algebraically.

Prove geometric theorems.

Make geometric constructions.

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

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

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

Analyze, compare, create, and compose shapes.

Apply geometric concepts in modeling situations.

Reason with shapes and their attributes.

Define trigonometric ratios and solve problems involving right triangles.

Understand congruence in terms of rigid motions.

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

Understand and apply the Pythagorean Theorem.

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

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

Apply trigonometry to general triangles.

Prove theorems involving similarity.

Understand and apply theorems about circles.

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

Find arc lengths and areas of sectors of circles.

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

Understand similarity in terms of similarity transformations.

Experiment with transformations in the plane.

Explain volume formulas and use them to solve problems.
problem_activity: Activity
This activity begins a sequence which looks at figures that are not polygons. From the point of view of congruence, polygons are special shapes because they are completely determined by the set of vertices. For curved shapes, we usually cannot check that they are congruent by examining a few privileged points, like the vertices of polygons. We
can
ascertain that they are not congruent by identifying a feature of one shape not shared by the other (for example, this oval is 3 units wide while this one is only
\(2 \frac{1}{2}\)
units wide). But to show that two curved shapes
are
congruent, we need to apply the definition of congruence and try to move one shape so that it matches up exactly with the other after some translations, rotations, and reflections.
In this activity, students begin to explore the subtleties of congruence for curved shapes. Make sure that students provide a solid mathematical argument for the shapes which are congruent, beyond saying that they look the same. Providing a viable argument, MP3, requires careful thinking about the meaning of congruence and the structure of the shapes. Monitor for groups who use precise language of transformations as they attempt to move one traced oval to match up perfectly with another. Invite them to share their reasoning during the discussion. Also monitor for arguments based on measurement for why neither of the upper ovals can be congruent to either of the lower ones.
Launch
Arrange students in groups of 2. Provide access to geometry toolkits. Give students 3 minutes of quiet work time, then invite them to share their reasoning with a partner, followed by a whole-class discussion.
Engagement: Develop Effort and Persistence
. Connect a new concept to one with which students have experienced success. For example, review the criteria used to determine congruence for polygons so that students can transfer these strategies in determining congruence for curved shapes.
Supports accessibility for: Social-emotional skills; Conceptual processing
Speaking: Math Language Routine 5 Co-Craft Questions.
This is the first time Math Language Routine 5 is suggested as a support in this course. In this routine, students are given a context or situation, often in the form of a problem stem with or without numerical values. Students develop mathematical questions that can be asked about the situation. A typical prompt is: “What mathematical questions could you ask about this situation?” The purpose of this routine is to allow students to make sense of a context before feeling pressure to produce answers, and to develop students’ awareness of the language used in mathematics problems.
Design Principle(s): Cultivating Conversations; Maximize meta-awareness
How It Happens:
Display the four images of the ovals without the directions.
Ask students, “What mathematical questions could you ask about this situation?”
Give students 1 minute of individual time to jot some notes, and then 3 minutes to share ideas with a partner.
As pairs discuss, support students in using conversation skills to generate and refine their questions collaboratively by seeking clarity, referring to students’ written notes, and revoicing oral responses, as necessary. Listen for how students use language about transformations and/or refer to measurements of the figures when talking about the curved polygons.
Ask each pair of students to contribute one written question to a poster, the whiteboard, or digital projection. Call on 2–3 pairs of students to present their question to the whole class, and invite the class to make comparisons among the questions shared and their own questions.
Listen for questions comparing different features of the ovals to determine congruence, especially those that use distances between corresponding points. Revoice student ideas with an emphasis on measurements wherever it serves to clarify a question.
Reveal the question, “Are any of the ovals congruent to one another?” and give students a couple of minutes to compare it to their own question and those of their classmates. Identify similarities and differences.
Consider providing these prompts: “Which of your questions is most similar to/different from the one provided? Why?”, “Is there a main mathematical concept that is present in both your questions and the one provided? If so, describe it.”, and “How do your questions relate to one of the lesson goals of measuring corresponding points to determine congruence?”
Invite students to choose one question to answer (from the class or from the curriculum), and then have students move on to the Activity Synthesis.
Student Facing
Are any of the ovals congruent to one another? Explain how you know.
[IMG]
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?
You can use 12 toothpicks to create a polygon with an area of five square toothpicks, like this:
Can you use exactly 12 toothpicks to create a polygon with an area of four square toothpicks?
[IMG]
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Activity Synthesis
Invite groups to explain how they determined that the upper ovals are not congruent to the lower ones, with at least one explanation focusing on differing measurable attributes (for example length and width). Also invite previously selected groups to show how they demonstrated that the two upper (and two lower) ovals are congruent, focusing on the precise language of transformations.
Emphasize that showing that two oval shapes are congruent
requires
using the definition of congruence: is it possible to move one shape so that it matches up perfectly with the other using only rigid transformations? Experimentation with transformations is essential when showing that two of the ovals match up because, unlike polygons, these shapes are not determined by a finite list of vertices and side lengths.
Students have seen that rectangles that have the same side lengths are congruent and will later find criteria for determining when two triangles are congruent. For more complex curved shapes, the definition of congruence is required.


correct_option_index: 6