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: Add and subtract within 20.

Perform arithmetic operations on polynomials.

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

Work with addition and subtraction equations.

Solve systems of equations.

Write and interpret numerical expressions.

Reason about and solve one-variable equations and inequalities.

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

Work with radicals and integer exponents.

Represent and solve equations and inequalities graphically.

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

Apply and extend previous understandings of arithmetic to algebraic expressions.

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

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

Solve equations and inequalities in one variable.

Analyze patterns and relationships.

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

Gain familiarity with factors and multiples.

Use the four operations with whole numbers to solve problems.

Interpret the structure of expressions.

Represent and solve problems involving addition and subtraction.

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

Understand the relationship between zeros and factors of polynomials.

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

Represent and solve problems involving multiplication and division.

Create equations that describe numbers or relationships

Use properties of operations to generate equivalent expressions.

Multiply and divide within 100.

Use polynomial identities to solve problems.

Generate and analyze patterns.

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

Rewrite rational expressions.

Represent and analyze quantitative relationships between dependent and independent variables.

Write expressions in equivalent forms to solve problems.
problem_activity: Activity
In the previous activity students looked at 9 different equations. In this activity, each student focuses on just one of those equations to learn about the real-world situation it represents. They interpret the variables and values in terms of the situation (MP7). Adding to the table they saw in the previous activity gives students an opportunity to practice both evaluating expressions and solving equations.
This activity also asks students to create a graph of their assigned relationship on graph paper. In previous activities, students were given a grid with the axes already labeled and numbered whenever they were asked to create a graph of a relationship. Here they must decide for themselves how to scale each axis.
Launch
Decide whether you want to assign the equations or allow students to select which equation they will work with. Make sure there is at least one student working with each equation.
Give students 5–10 minutes of quiet work time followed by partner and whole-class discussion. If there is more than one student working with the same equation, consider letting them share their answers and reasoning with each other, after they have had some time to work on their own.
Action and Expression: Internalize Executive Functions.
Chunk this task into more manageable parts. Check in with students within the first 2-3 minutes of work time. Look for students who need additional support deciding how to scale each axis for table. Invite students to share strategies they have used so far, as well as any questions they have before continuing.
Supports accessibility for: Memory; Organization
Student Facing
The equations in the previous activity represent situations.
\(S - 2 = T\)
where
\(S\)
is the number of sides on a polygon and
\(T\)
is the number of triangles you can draw inside it (from one vertex to the others, without overlapping)
\(G = J + 13\)
where
\(G\)
is a day in the Gregorian calendar and
\(J\)
is the same day in the Julian calendar
\(P = I - 47.50\)
where
\(I\)
is the amount of income and
\(P\)
is the profit after
$
47.50 in expenses
\(C + 273.15 = K\)
where
\(C\)
is a temperature in degrees Celsius and
\(K\)
is the same temperature in Kelvin
\(e = 6s\)
where
\(e\)
is the total edge length of a regular tetrahedron and
\(s\)
is the length of one side
\(m = 8.96V\)
where
\(V\)
is the volume of a piece of copper and
\(m\)
is its mass
\(y = \frac{1}{12}x\)
where
\(x\)
is the number of eggs and
\(y\)
is how many dozens that makes
\(t = \frac{d}{2.5}\)
where
\(t\)
is the amount of time it takes in seconds to jog a distance of
\(d\)
meters at a constant speed of 2.5 meters per second
\(g = 28.35z\)
where
\(g\)
is the mass in grams and
\(z\)
is the same amount in ounces
Your teacher will assign you one of these equations to examine more closely.
Rewrite your equation using words. Use words like product, sum, difference, quotient, and term.
In the previous activity, you matched equations and tables. Copy the values from the table that matched your assigned equation into the first 3 rows of this table. Make sure to label what each column represents.
| independent variable:                             ________________________ | dependent variable:                             ________________________ |
|----------------------------------------------------------------------------|--------------------------------------------------------------------------|
|                                                                            |                                                                          |
|                                                                            |                                                                          |
|                                                                            |                                                                          |
|                                     60                                     |                                                                          |
|                                                                            |                                   300                                    |
Select one of the first 3 rows of the table and explain what those values mean in this situation.
Use your equation to find the values that complete the last 2 rows of the table. Explain your reasoning.
On graph paper, create a graph that represents this relationship. Make sure to label your axes.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Anticipated Misconceptions
Some students may struggle to understand what each step is asking them to do. Consider preparing an example to share with them, such as this for the equation
\(2.54i = c\)
, and point out which part of the example corresponds to the question they are on.
The number of centimeters is the product of the number of inches and 2.54.
| independent variable:                       length (inches) | dependent variable:                       length (centimeters) |
|-------------------------------------------------------------|----------------------------------------------------------------|
|                              5                              |                              12.7                              |
|                             36                              |                             91.44                              |
|                             75                              |                             190.5                              |
|                             60                              |                                                                |
|                                                             |                              300                               |
If something is 36 inches long, this is equivalent to 91.44 when measured in centimeters.
In the table above, fill in 152.4 in the right column, next to the 60, because
\(2.54 \boldcdot 60 = 152.4\)
. Fill in 118.11 in the left column, next to the 300, because
\(2.54 \boldcdot 118.11 \approx 300\)
A graph with “length (inches)” on the horizontal axis, “length (centimeters)” on the vertical axis, and points at
\((5, 12.7), (36, 91.44), (75, 190.5), (60, 152.4)\)
, and
\((118.11, 300)\)
Some students may struggle with setting up their graph from scratch on graph paper. Prompt them to think about the maximum value they want to represent on each axis and what number they could count by to get to that maximum value in the amount of space they have.
Activity Synthesis
Continue to the next activity. The discussion of these relationships occurs after students have created their visual displays.


correct_option_index: 6

11

32