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 and solve linear equations and pairs of simultaneous 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.

Understand the relationship between zeros and factors of polynomials.

Use polynomial identities to solve problems.

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

Rewrite rational expressions.

Work with radicals and integer exponents.

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

Represent and solve problems involving addition and subtraction.

Work with addition and subtraction equations.

Represent and solve problems involving multiplication and division.

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

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

Reason about and solve one-variable equations and inequalities.

Solve systems of equations.

Multiply and divide within 100.

Write expressions in equivalent forms to solve problems.

Create equations that describe numbers or relationships

Perform arithmetic operations on polynomials.

Analyze patterns and relationships.

Solve equations and inequalities in one variable.

Add and subtract within 20.

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

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

Represent and analyze quantitative relationships between dependent and independent variables.

Interpret the structure of expressions.

Write and interpret numerical expressions.

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

Gain familiarity with factors and multiples.

Represent and solve equations and inequalities graphically.

Use the four operations with whole numbers to solve problems.

Apply and extend previous understandings of arithmetic to algebraic expressions.

Generate and analyze patterns.
problem_activity: Narrative
In this activity, students solve problems that involve finding multiples that are shared by two different numbers: the number of hot dogs in a package and the number of hot dog buns in a package. They reason about how many of each package to get to make a certain number of servings of hot dogs. As multiple answers can be expected, the focus is on explaining why the solutions make sense (MP3).
To solve the problems, students may decontextualize the situation and reason about factors and multiples, and then recontextualize the solutions in terms of servings of hot dogs. As they do so, they practice reasoning quantitatively and abstractly (MP2). During the synthesis, analyze different solutions and discuss why numbers that are multiples both of 8 and 10 are useful in this situation.
Launch
Groups of 2–4
Read the opening paragraph as a class.
“What questions could you ask about the situation?”
1 minute: quiet think time
Share responses.
“Let’s solve problems about hot dogs and hot dog buns, which come in different-size packages.”
“For each problem, look for possible numbers of packages that are needed to serve hot dogs at a picnic.”
“Take a minute to read the problems and think about how you would solve them.”
1 minute: quiet think time
Activity
“Now work with your group to solve these problems.”
8–10 minutes: group work time
Monitor for students who:
use the term “multiple” in their explanations
try different possibilities for the number of packages to buy
try to find numbers of packages that would give the exact (or close to the exact) numbers of servings
Student Facing
Each package of hot dogs has 10 hot dogs. Each package of hot dog buns has 8 buns.
Lin expects to need 50 hot dogs for a class picnic.
How many packages of hot dogs should Lin get? Explain or show your reasoning.
Can Lin get exactly 50 hot dog buns? How many packages of hot dog buns should Lin get? Explain or show your reasoning.
Diego expects to need 72 hot dogs for a class picnic.
How many packages of hot dogs should Diego get? Explain or show your reasoning.
How many packages of hot dog buns should Diego get? Explain or show your reasoning.
Is it possible to buy exactly the same number of hot dogs and buns? If you think so, what would that number be? If not, explain your reasoning.
Student Response
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Advancing Student Thinking
If students are using guess and check, consider asking: “How are you choosing which numbers to try?” or “How can thinking about multiples help you solve the problem?”
Activity Synthesis
Invite students to share their responses to the problem about 72 hot dogs, including those who get 70 hot dogs and 72 buns or 80 hot dogs and 80 buns. Encourage students to highlight how they are using multiples in their reasoning.
“Why are there different answers to this question?” (The numbers do not work out exactly to get 72 hot dogs and 72 buns, so we need to look at different alternatives.)
“Is it possible to buy exactly the same number of hot dogs and buns? If so, how many of each?” (Yes, 40, 80.)
“Why is it possible to buy exactly the same number of hot dogs and buns when the packaging is different?” (Some multiples of 10 are also multiples of 8, such as 40 and 80.)


correct_option_index: 29

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