few_shots: Example:
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_001392

8
options: Compute fluently with multi-digit numbers and find common factors and multiples.

Represent and model with vector quantities.

Reason quantitatively and use units to solve problems.

Use properties of rational and irrational numbers.

Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.

Extend the properties of exponents to rational exponents.

Perform operations on vectors.

Use complex numbers in polynomial identities and equations.

Represent complex numbers and their operations on the complex plane.

Perform operations on matrices and use matrices in applications.

Know that there are numbers that are not rational, and approximate them by rational numbers.

Apply and extend previous understandings of multiplication and division to divide fractions by fractions.

Perform arithmetic operations with complex numbers.

Apply and extend previous understandings of numbers to the system of rational numbers.
problem_activity: Activity
The purpose of this activity is for students to use the four operations on rational numbers solve a problem about water in a tank. The activity presents another example where negative time is used; this time to describe before a sensor starts working. Students examine the change as a separate column before using the starting point to model the draining of the tank.
Students who see that they are doing the same computations over and over and see that the structure of the expression is the same every time are expressing regularity in repeated reasoning (MP8). Monitor for different explanations for the last question.
Launch
Arrange students in groups of 2. Give students 4 minutes of quiet work time, followed by partner and whole-class discussion.
Action and Expression: Internalize Executive Functions.
Chunk this task into more manageable parts. For example, after students have completed the first 2-3 rows of the first table, check-in with either select groups of students or the whole class. Invite students to share how they have applied generalizations from previous lessons about using the four operations on rational numbers so far.
Supports accessibility for: Conceptual processing; Organization; Memory
Student Facing
A tank of water is being drained. Due to a problem, the sensor does not start working until some time into the draining process. The sensor starts its recording at time zero when there are 770 liters in the tank.
Given that the drain empties the tank at a constant rate of 14 liters per minute, complete the table:
| time after sensor                       starts (minutes) | change in                       water (liters) |         expression         | water in the                       tank (liters) |
|----------------------------------------------------------|------------------------------------------------|----------------------------|--------------------------------------------------|
|                            0                             |                       0                        |  \(770 + (0) (\text-14)\)  |                       770                        |
|                            1                             |                      -14                       | \(770 + (1)  (\text-14) \) |                       756                        |
|                            5                             |                      -70                       |                            |                                                  |
|                            10                            |                                                |                            |                                                  |
Later, someone wants to use the data to find out how long the tank had been draining before the sensor started. Complete this table:
| time after sensor                       starts (minutes) | change in                       water (liters) |            expression            | water in the                       tank (liters) |
|----------------------------------------------------------|------------------------------------------------|----------------------------------|--------------------------------------------------|
|                            1                             |                      -14                       |    \(770 + (1) (\text-14) \)     |                       756                        |
|                            0                             |                       0                        |    \(770 + (0)  (\text-14) \)    |                       770                        |
|                            -1                            |                       14                       | \(770 + (\text-1)  (\text-14) \) |                       784                        |
|                            -2                            |                       28                       |                                  |                                                  |
|                            -3                            |                                                |                                  |                                                  |
|                            -4                            |                                                |                                  |                                                  |
|                            -5                            |                                                |                                  |                                                  |
If the sensor started working 15 minutes into the tank draining, how much was in the tank to begin with?
Student Response
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Anticipated Misconceptions
The last question might be hard for students because they have had the table to support calculating the answers.  Ask students how can they use the previous entries in the table to help them calcualte the answer?  What is something they see changing in the expressions in the table that would change in this question?
Activity Synthesis
Make sure students have filled out the tables appropriately. Select students to share their reasoning for the last one.
Speaking, Representing: MLR7 Compare and Connect.
Use this routine when students share their strategies for completing the table. Ask students to consider what is the same and what is different about the structure of each expression. Draw students’ attention to the connection between representations (e.g., “Where do you see the change in water in your expression?”, "How are do repeated computations appear in the expression?"). These exchanges strengthen students’ mathematical language use and reasoning with different representations.
Design Principle(s): Support sense-making; Maximize meta-awareness


correct_option_index: 4