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: Work with radicals and integer exponents.

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

Write and interpret numerical expressions.

Generate and analyze patterns.

Perform arithmetic operations on polynomials.

Solve systems of equations.

Reason about and solve one-variable equations and inequalities.

Gain familiarity with factors and multiples.

Apply and extend previous understandings of arithmetic to algebraic expressions.

Multiply and divide within 100.

Use polynomial identities to solve problems.

Rewrite rational expressions.

Represent and analyze quantitative relationships between dependent and independent variables.

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

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

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

Add and subtract within 20.

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

Represent and solve problems involving addition and subtraction.

Work with addition and subtraction equations.

Create equations that describe numbers or relationships

Understand the relationship between zeros and factors of polynomials.

Solve equations and inequalities in one variable.

Use properties of operations to generate equivalent expressions.

Represent and solve equations and inequalities graphically.

Write expressions in equivalent forms to solve problems.

Use the four operations with whole numbers to solve problems.

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

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

Analyze patterns and relationships.

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

Represent and solve problems involving multiplication and division.

Interpret the structure of expressions.

Understand the connections between proportional relationships, lines, and linear equations.
problem_activity: Activity
The purpose of this activity is to give students an opportunity to practice checking whether a particular value is a solution to an equation, and to recall properties of operations and equality that preserve the solution set of an equation.
This will be useful when students consider when and why equations have the same solution in the associated Algebra 1 lesson.
Monitor for students who:
check the value of
\(x\)
in only one expression and reason about the second expression
identify properties by name such as commutative, distributive, and associative
identify properties informally such as “changing the order doesn’t matter” or “you can add first and then multiply or multiply first and then add” or “you can multiply three or more terms in any order” or “you can add the same thing to each side”
manipulate the expressions
graph one or both expressions
Making graphing technology available gives students an opportunity to choose appropriate tools strategically.
Student Facing
For each pair of equations, decide whether the given value of
\(x\)
is a solution to one or both equations:
Is
\(x = 2\)
a solution to:
\(x(2 + 3) = 10\)
\(2x + 3x = 10\)
Is
\(x = 3\)
a solution to:
\(x - 4 = 1\)
\(4 - x = 1\)
Is
\(x = \text{-}2\)
a solution to:
\(7x = \text{-}14\)
\(x \boldcdot 14 = \text{-}28\)
Is
\(x = \text{-}1\)
a solution to:
\(x + 3 = 2\)
\(3 + x = 2\)
Is
\(x = \text{-5}\)
a solution to:
\(3 - x = 8\)
\(5 - x = 10\)
Is
\(x = (8 + 1)+3\)
a solution to:
\(\frac{12}{2} = \frac12(x)\)
\(18 = 2x\)
Is
\(x = 2\)
a solution to:
\(\frac{12}{x} = 6\)
\(6x = 12\)
Is
\(x = \frac{10}{3}\)
a solution to:
\(\text{-}1 + 3x = 9\)
\(9 = 3x - 1\)
Is
\(x = \frac12\)
a solution to:
\(5(x + 1) = \frac{15}{2}\)
\(5x + 1 = \frac{15}{2}\)
Student Response
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for free access to Student Response.
Activity Synthesis
Ask students to talk to their neighbor about some things they can look for, to see if two equations are likely to have the same solution or different solutions. If needed, draw students’ attention to which equations had the same solution and which had different solutions, perhaps by displaying them for all to see, grouped by whether the equations were equivalent or not.
Call on students to share their thinking and make an informal list displayed for all to see. It’s fine if students do not name all possible properties, or only describe them informally. They will have more opportunities to encounter them formally in their Algebra 1 course.
Remind students that
equivalent
equations have the same solution set. Use the language of
equivalent
interchangeably with “have the same solution,” and monitor for students to pick up on this vocabulary.


correct_option_index: 8