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: Analyze functions using different representations.

Understand the concept of a function and use function notation.

Model periodic phenomena with trigonometric functions.

Interpret expressions for functions in terms of the situation they model.

Prove and apply trigonometric identities.

Construct and compare linear and exponential models and solve problems.

Extend the domain of trigonometric functions using the unit circle.

Define, evaluate, and compare functions.

Build new functions from existing functions.

Build a function that models a relationship between two quantities.

Interpret functions that arise in applications in terms of the context.

Use functions to model relationships between quantities.
problem_activity: Activity
In this warm-up, students considers what happens if an object is launched up in the air unaffected by gravity. The work here serves two purposes. It reminds students that an object that travels at a constant speed can be described with a linear function. It also familiarizes students with a projectile context used in the next activity, in which students will investigate a quadratic function that more realistically models the movement of a projectile—with gravity in play.
Students who use a spreadsheet to complete the table practice choosing tools strategically (MP5).
Launch
Ask a student to read the opening paragraph of the activity aloud. To help students visualize the situation described, consider sketching a picture of a cannon pointing straight up, 10 feet above ground. Ask students to consider what a speed of 406 feet per second means in more concrete terms. How fast is that?
Students may be more familiar with miles per hour. Tell students that the speed of 406 feet per second is about 277 miles per hour.
Consider arranging students in groups of 2 so they can divide up the calculations needed to complete the table. Provide access to calculators, if requested.
Student Facing
A cannon is 10 feet off the ground. It launches a cannonball straight up with a velocity of 406 feet per second.
Imagine that there is no gravity and that the cannonball continues to travel upward with the same velocity.
Complete the table with the heights of the cannonball at different times.
|           seconds            |   |   |   |   |   |   |
|------------------------------|---|---|---|---|---|---|
| distance above ground (feet) |   |   |   |   |   |   |
Write an equation to model the distance in feet,
\(d\)
, of the ball
\(t\)
seconds after it was fired from the cannon if there was no gravity.
Student Response
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Activity Synthesis
Ask students how the values in the table are changing and what equation would describe the height of the cannonball if there were no gravity. Even without graphing, students should notice that the height of the cannonball over time is a linear function given the repeated addition of 406 feet every time
\(t\)
increases by 1.
Tell students that, in the next activity, they will look at some actual heights of the cannonball.


correct_option_index: 9