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: Extend the counting sequence.

Generalize place value understanding for multi-digit whole numbers.
Grade 4 expectations in this domain are limited to whole numbers less than or equal to 1,000,000.

Use place value understanding and properties of operations to add and subtract.

Understand the place value system.

Perform operations with multi-digit whole numbers and with decimals to hundredths.

Use place value understanding and properties of operations to perform multi-digit arithmetic. (A range of algorithms may be used.)

Use place value understanding and properties of operations to perform multi-digit arithmetic.
Grade 4 expectations in this domain are limited to whole numbers less than or equal to 1,000,000.

Work with numbers 11–19 to gain foundations for place value.

Understand place value.
problem_activity: Narrative
The purpose of this activity is for students to compare three-digit numbers based on their understanding of place value. They are invited to explain or show their thinking in any way that makes sense to them. A number line is provided. Students may revise their thinking after locating the numbers on the number line, or may choose to draw diagrams to represent their thinking. During the activity synthesis, methods based on comparing the value of digits by place are highlighted.
For the last problem, students persevere in problem solving as there are many ways to make most of inequalities true but students will need to think strategically in order to fill out all of them. In particular, 810 can be used in the first, second, or fourth inequality but it needs to be used in the fourth because it is the
only
number on the list that is larger than 793.
Representation: Develop Language and Symbols.
Represent the problem in multiple ways to support understanding of the situation. For example, have one group member place the numbers on a number line to verify which number is greater. Another option is to have one group member make the numbers using base-ten blocks to prove which is greater.
Supports accessibility for: Conceptual Processing, Organization
Launch
Groups of 2
“In the warm-up, you saw that different forms of writing a number can help you think about the value of each digit.”
Write
\(564\phantom{3} \boxed{\phantom{33}}\phantom{3}504\)
on the board.
“What symbol would make this expression true? Explain.” (>, because they both have 500, but the first number has 6 tens or 60 and the second number has 0 tens.)
Activity
“Today you will be comparing three-digit numbers by looking at place value.”
“If it helps, you can use base-ten diagrams or expanded form to help you think about place value.”
“Try it on your own and then compare with your partner.”
5 minutes: independent work time
5 minutes: partner work time
Monitor for students who compare the numbers without drawing base-ten diagrams or using a number line.
Student Facing
Compare the numbers.
\(>\)
,
\(=\)
, or
\(<\)
521
\(\boxed{\phantom{33}}\)
523
Explain or show your thinking. If it helps, use a diagram or number line.
\(>\)
,
\(=\)
, or
\(<\)
889
\(\boxed{\phantom{33}}\)
878
Explain or show your thinking. If it helps, use a diagram or number line.
Place the numbers in the blanks to make each comparison true. Use each number only once. Use base-ten diagrams or a number line if it helps.
810
529
752
495
\(\underline{\hspace{1 cm}} > 519\)
\(687 < \underline{\hspace{1 cm}}\)
\(\underline{\hspace{1 cm}} < 501\)
\(\underline{\hspace{1 cm}} > 793\)
Student Response
Teachers with a valid work email address can
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Advancing Student Thinking
If students write comparison statements that are not true, consider asking:
“Could you read each statement aloud?”
“How did you know your statement is true?”
“How could you use the base-ten diagram or number line to help you show whether your statement is true or false?”
Activity Synthesis
Display
\(564 > 504\)
.
Display
\(500 + 60 + 4 > 500 + 4\)
.
“We decided that this was a true statement. How does the expanded form of these numbers help justify our thinking?” (We can see the value of each place, so we can compare each digit.)
Invite previously selected students to share how they compared numbers without drawing diagrams or using the number line.


correct_option_index: 8