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

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

Understand and apply the Pythagorean Theorem.

Prove geometric theorems.

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

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

Understand congruence in terms of rigid motions.

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

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

Reason with shapes and their attributes.

Explain volume formulas and use them to solve problems.

Understand similarity in terms of similarity transformations.

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

Analyze, compare, create, and compose shapes.

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

Make geometric constructions.

Experiment with transformations in the plane.

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

Define trigonometric ratios and solve problems involving right triangles.

Find arc lengths and areas of sectors of circles.

Understand and apply theorems about circles.

Apply geometric concepts in modeling situations.

Prove theorems involving similarity.

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

Apply trigonometry to general triangles.

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

Use coordinates to prove simple geometric theorems algebraically.
problem_activity: Narrative
The purpose of this activity is for students to identify and name shapes in the environment. The shape walk can take place in the playground, library, or another area of the school. It may be easier for students to use their recording sheet if it is on a clipboard.
When students recognize mathematical features of objects in the real world, they model with mathematics (MP4).
Engagement: Develop Effort and Persistence.
Students may benefit from chunking this task into smaller parts. Invite students to look for one shape at a time. Check in with students to provide feedback and encouragement after recording each shape.
Supports accessibility for: Conceptual Processing
Required Materials
Materials to Gather
Clipboards
Launch
Groups of 2
“We are going to take a walk around the playground. Look for examples of triangles, rectangles, and circles around the playground. Tell your partner about the shapes you find and draw pictures or write words to record what you find.”
Activity
10 minutes: shape walk
“Write a number to show how many of each shape you found.”
2 minutes: independent work time
Student Facing
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Synthesis:
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Student Response
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for free access to Student Response.
Activity Synthesis
Invite students to share their recording sheets and the shapes that they found.
Display the image or an example from the playground:
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“Han found this swing on the playground. Where should he draw this on the recording sheet?” (The swing looks like a rectangle so he could draw it with the rectangles. The sides of the swing look like triangles.)


correct_option_index: 7

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