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:
{
  "explanation": "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:
{
  "explanation": "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.
{"table": [{"_attributes": {"align": "center", "class": ["table", "table-bordered"], "style": "table-layout:fixed; width:70%"}, "tbody": [{"tr": [{"td": [{"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}, "strong": [{"_value": "No solution"}]}, {"_attributes": {"style": "text-align:center"}, "strong": [{"_value": "Exactly 1 solution"}]}, {"_attributes": {"style": "text-align:center"}, "strong": [{"_value": "Infinite number of solutions"}]}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${9=4x+7}$$"}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${y=3y+5}$$"}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${y={x\\over2}}$$"}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"p": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${3x+6y-9=0}$$"}]}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"p": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${x=y}$$"}]}, {"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${1.25x=1.25y}$$"}]}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"p": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${9.9=6x+8y}$$"}]}, {"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${9x=2.5y-8.8}$$"}]}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"p": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${11x-2y=1.5}$$"}]}, {"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${11x-2y=2.5}$$"}]}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {}]}]}]}]}
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:
{
  "explanation": "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: Work with radicals and integer exponents.

Solve problems involving the four operations, and identify and explain patterns in arithmetic.

Apply and extend previous understandings of arithmetic to algebraic expressions.

Analyze and solve linear equations and pairs of simultaneous linear equations.

Understand properties of multiplication and the relationship between multiplication and division.

Interpret the structure of expressions.

Gain familiarity with factors and multiples.

Rewrite rational expressions.

Understand solving equations as a process of reasoning and explain the reasoning.

Use polynomial identities to solve problems.

Add and subtract within 20.

Write and interpret numerical expressions.

Understand the connections between proportional relationships, lines, and linear equations.

Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.

Use properties of operations to generate equivalent expressions.

Perform arithmetic operations on polynomials.

Solve real-life and mathematical problems using numerical and algebraic expressions and equations.

Multiply and divide within 100.

Work with equal groups of objects to gain foundations for multiplication.

Understand the relationship between zeros and factors of polynomials.

Use the four operations with whole numbers to solve problems.

Solve equations and inequalities in one variable.

Understand and apply properties of operations and the relationship between addition and subtraction.

Reason about and solve one-variable equations and inequalities.

Solve systems of equations.

Write expressions in equivalent forms to solve problems.

Represent and analyze quantitative relationships between dependent and independent variables.

Represent and solve equations and inequalities graphically.

Work with addition and subtraction equations.

Represent and solve problems involving multiplication and division.

Generate and analyze patterns.

Analyze patterns and relationships.

Represent and solve problems involving addition and subtraction.

Create equations that describe numbers or relationships
problem_activity: Activity
One purpose of this activity is to practice seeing the total length of a segment as the sum of its pieces, and using the whole side length of a rectangle to express its area. Another is to generate examples of equivalent expressions, and understand why they are equivalent based on an understanding of area and the distributive property.
Monitor for students who write different but equivalent expressions to represent the area of a rectangle. For example,
\(2(t+5+t)\)
,
\(2t+10+2t\)
, and
\(4t+10\)
for figure B.
Student Facing
Complete the table with the length, width, and area of each rectangle.
[IMG]
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}, "_value": "rectangle"}, {"_attributes": {"scope": "col"}, "_value": "length (units)"}, {"_attributes": {"scope": "col"}, "_value": "width (units)"}, {"_attributes": {"scope": "col"}, "_value": "area (square units)"}]}]}], "tbody": [{"tr": [{"td": [{"_value": "A"}, {"span": [{"_attributes": {"class": ["math", "math-repaired"]}, "_value": "\\(a+4\\)"}]}, {}, {}]}, {"td": [{"_value": "B"}, {}, {"_value": "2"}, {}]}, {"td": [{"_value": "C"}, {}, {}, {}]}, {"td": [{"_value": "D"}, {}, {}, {}]}, {"td": [{"_value": "E"}, {}, {}, {}]}]}]}]}
Student Response
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for free access to Student Response.
Activity Synthesis
Invite previously-selected students to share different but equivalent expressions for length and area. (Reassure students that it doesn’t matter which side is considered the length or the width.) If any students express confusion over a response, encourage the student who came up with the response to explain. For example, a student might wonder why
\(a+a+a\)
can be expressed as
\(3a\)
.
Display the table from the task statement for all to see, and write any equivalent expressions that are mentioned next to each other. For example, for the area of figure A, you might write
\(5(a+4)\)
,
\((4+a) \boldcdot 5\)
, and
\(5a+20\)
. Show in the figure how you can see
\(5(a+4)\)
(the product of one side and the other whole side) and
\(5a+20\)
(the area of each smaller rectangle, added up). Emphasize that all of these expressions correctly represent the area of the whole figure, and highlight use of the distributive property where it occurs.


correct_option_index: 2