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: Describe and compare measurable attributes.

Tell and write time.

Measure and estimate lengths in standard units.

Geometric measurement: understand concepts of area and relate area to multiplication and to addition.

Represent and interpret data.

Geometric measurement: understand concepts of angle and measure angles.

Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects.

Work with time and money.

Convert like measurement units within a given measurement system.

Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit.

Relate addition and subtraction to length.

Classify objects and count the number of objects in each category.

Measure lengths indirectly and by iterating length units.

Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area measures.

Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition.
problem_activity: Narrative
The purpose of this activity is for students to identify objects that are longer or shorter than a given object. Students find two objects that are longer and two objects that are shorter than an unsharpened pencil. The students use the language of “longer than” and “shorter than” and record their findings in complete sentences so that indirect comparison can be discussed during the activity synthesis.
MLR8 Discussion Supports.
Think aloud and use gestures to emphasize the language used for comparison. For example, use hands further or closer apart to accompany the words “longer” and “shorter”.
Advances: Listening, Representing
Required Materials
Materials to Gather
Materials from a previous lesson
Pencils
Required Preparation
Each group of 4 students needs access to the materials from the previous lesson and one unsharpened pencil.
Create a two-column chart with the headings “longer” and “shorter” for the synthesis.
Launch
Groups of 4
Give each group an unsharpened pencil and access to the objects from the previous lesson.
Display two objects that are not the same length (for example, a crayon and a marker).
“How do the lengths of these objects compare?” (The crayon is shorter than the marker. The marker is longer than the crayon.)
30 seconds: quiet think time
Share responses.
Activity
“You are going to compare the length of objects to the length of the unsharpened pencil. Find two objects that are longer than the pencil and two that are shorter than the pencil. Write a sentence for each comparison.”
8 minutes: independent work time
Monitor for two objects that are close in length but one is found to be longer than the unsharpened pencil and one is found to be shorter.
Student Facing
Find 2 objects that are longer and 2 objects that are shorter than the unsharpened pencil.
Write a sentence that compares each object to the pencil.
Example: The bulletin board
is longer than the pencil.
Student Response
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Activity Synthesis
Display the T-chart with headings “longer” and “shorter.”
“What objects are longer than an unsharpened pencil?” (a book, the ruler, the height of the desk)
Record responses.
What objects are shorter than an unsharpened pencil?” (a marker cap, a crayon, a toothpick)
Record responses.
Display previously identified objects that are close in length.
“How do the lengths of the two objects compare?” (The book that is shorter than the pencil is also shorter than the block because the block is longer than the pencil.)
“Because we compared the length of objects to the pencil, we know that some objects are longer or shorter than others without lining them up.”


correct_option_index: 12