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 proportional relationships and use them to solve real-world and mathematical problems.

Understand ratio concepts and use ratio reasoning to solve problems.
problem_activity: Optional activity
This activity is the same type of situation as the previous one: comparing the voting of two groups on a yes or no issue. However, the numbers make it more difficult to use "part to part" ratios. Again, students need to be thinking about how to make sense of (MP1) and quantify the class voting decisions (MP2).
Launch
Arrange students in groups of 2–4. Provide access to four-function calculators.
Student Facing
The school will be painted over the summer. Students get to vote on whether to change the color to purple (a “yes” vote), or keep it a beige color (a “no” vote).
The principal of the school decided to analyze voting results by class. The table shows some results.
In both classes, a majority voted for changing the paint color to purple. Which class was more in favor of changing?
|         | yes | no |
|---------|-----|----|
| class A |     |    |
| class B |     |    |
Student Response
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Anticipated Misconceptions
If students do not understand what comparing the two classes means, give a more extreme example, such as comparing 20 to 1 with 11 to 10. The 20 to 1 class is much more purple because almost everyone said yes. The other class had almost equal yesses and nos.
Students may be stuck with the difficult-looking numbers, expecting to be able to do calculations to create equivalent ratios mentally. Suggest that they find the total number of votes in each class.
Students may compute percentages incorrectly, forgetting that percentages are rates out of 100. So it is
not
correct to say that room A has 54% yesses. However, you can carefully make sense of this percent as a comparison: Class A had 54% as many nos as yesses; Class B had 61% as many nos as yesses. This means that Class B was more no-y, so it was less yessy than Class A.
Activity Synthesis
Invite students to show different ways of solving the problem, including using equivalent ratios and percentages. Ask students to explain their thinking. Correct ideas with incorrect calculations are still worth sharing.
Representation: Develop Language and Symbols.
Display or provide charts with symbols and meanings. Once students have determined which class was more in favor of changing, pause the class. Invite students to share their strategies for finding each fraction using percents and equivalent ratios to justify their reasoning. Create a display that includes each strategy labeled with the name and the fraction they represent of a total. Keep this display visible as students move on to the next problems.
Supports accessibility for: Conceptual processing; Memory


correct_option_index: 1