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.
{"table": [{"_attributes": {"align": "center", "class": ["table", "table-bordered"], "style": "table-layout:fixed; width:70%"}, "tbody": [{"tr": [{"td": [{"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}, "strong": [{"_value": "No solution"}]}, {"_attributes": {"style": "text-align:center"}, "strong": [{"_value": "Exactly 1 solution"}]}, {"_attributes": {"style": "text-align:center"}, "strong": [{"_value": "Infinite number of solutions"}]}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${9=4x+7}$$"}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${y=3y+5}$$"}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${y={x\\over2}}$$"}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"p": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${3x+6y-9=0}$$"}]}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"p": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${x=y}$$"}]}, {"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${1.25x=1.25y}$$"}]}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"p": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${9.9=6x+8y}$$"}]}, {"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${9x=2.5y-8.8}$$"}]}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"p": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${11x-2y=1.5}$$"}]}, {"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${11x-2y=2.5}$$"}]}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {}]}]}]}]}
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:
{
  "explanation": "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: Build a function that models a relationship between two quantities.

Build new functions from existing functions.

Define, evaluate, and compare functions.

Construct and compare linear and exponential models and solve problems.

Model periodic phenomena with trigonometric functions.

Interpret functions that arise in applications in terms of the context.

Understand the concept of a function and use function notation.

Interpret expressions for functions in terms of the situation they model.

Extend the domain of trigonometric functions using the unit circle.

Prove and apply trigonometric identities.

Analyze functions using different representations.

Use functions to model relationships between quantities.
problem_activity: Activity
This activity serves as a brief and light introduction to how
\(e\)
is visible in the behaviors of certain functions. The standards addressed in this course don’t require a deep understanding of the meaning of
\(e\)
, so the observations from this activity should not be assessed.
\(e\)
has a special connection to the expression
\(\left(1+\frac{1}{x}\right)^x\)
. Jumping to this expression would likely be an opaque process for most students, so they first investigate the range of behavior of its component pieces, for very small and very large positive values of
\(x\)
.
Entering increasingly larger or smaller values of
\(x\)
in the functions and making observations about them is a chance to look for and express regularity through repeated reasoning (MP8).
Launch
Tell students that they are to analyze the behavior of some functions and see what observations could be made about
\(e\)
along the way. Provide access to graphing technology and spreadsheet tools.
Arrange students in groups of 2. Tell students that they each should analyse the functions in different ways and then compare results. For example, one student could examine the functions numerically, using a table or a spreadsheet, while another student could examine them graphically.
Consider naming some tiny positive values of
\(x\)
(for example, 0.1, 0.01, and 0.001) and large values of
\(x\)
(for example, 100, 1,000, and 100,000) to students who may benefit from concrete examples.
Action and Expression: Internalize Executive Functions.
Provide students with a graphic organizer to capture their observations. Provide seven columns for students to organize their work. The first column label should be “function,” while the remaining six columns are divided into two sets; three for the three “tiny input” numbers and three for the “large input numbers.” Ask students to select three tiny input numbers and three large input numbers (either from the numbers given, or their own). Demonstrate sequencing the numbers for the column headers from left to right getting tinier, and from left to right getting larger. Explain to students that some may not need all six columns to draw conclusions. Students may especially want to capture observations and calculations for all six input values for
\(f(x)\)
and
\(g(x)\)
.
Supports accessibility for: Language; Organization
Student Facing
Here are some functions. For each function, describe, in words, the outputs for very tiny, positive values of
\(x\)
and for very large values of
\(x\)
.
\(a(x) = 1^x\)
\(b(x) = \text-x\)
\(d(x) = \frac{1}{x}\)
\(f(x) = \left(\frac{1}{x}\right)^x\)
\(g(x) = \left(1 + \frac{1}{x}\right)^x\)
\(h(x) = e^{x}\)
\(k(x) = 1 + x\)
Remember that
\(e \approx 2.718\)
. What does the function
\(g\)
have to do with the number
\(e\)
?
What do you notice about the relationship between
\(h\)
and
\(k\)
for very small, positive values of
\(x\)
?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Student Facing
Are you ready for more?
Complete the table to show the value of each expression to the nearest hundred-thousandth. Two entries have already been completed as an example.
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "row"}, "span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "50"}, "_value": "\\(x\\)"}]}, {"_attributes": {"scope": "col"}, "span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "1139"}, "_value": "\\(2^x\\)"}]}, {"_attributes": {"scope": "col"}, "span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "1186"}, "_value": "\\(e^x\\)"}]}, {"_attributes": {"scope": "col"}, "span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "6640"}, "_value": "\\(3^x\\)"}]}]}]}], "tbody": [{"tr": [{"td": [{"_value": "0.1"}, {"_value": "1.07177"}, {"_value": "1.10517"}, {"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "26857"}, "_value": "\\(\\phantom{1.11612}\\)"}]}]}, {"td": [{"_value": "0.01"}, {}, {}, {}]}, {"td": [{"_value": "0.001"}, {}, {}, {}]}, {"td": [{"_value": "0.0001"}, {}, {}, {}]}]}]}]}
What do you notice about the values in the table?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Anticipated Misconceptions
Students may substitute
\(0\)
for
\(x\)
in order to see what happens for small positive values of
\(x\)
. Focus their attention on functions
\(d\)
,
\(f\)
, and
\(g\)
where the
\(0\)
does not belong to the domain. For the other functions, encourage students to look at a graph (using technology for example). Does the value when
\(x = 0\)
help figure out what values each function takes for small positive values of
\(x\)
?
Activity Synthesis
Invite students to share their observations for the first question and the supporting work (a table, a spreadsheet, or graphs) that led to these observations, if available.
Focus the conversation on the responses to the last two questions. If not already observed by students, highlight that:
The value of function
\(g\)
gets closer and closer to
\(e\)
as
\(x\)
gets larger and larger. Display a graph representing
\(g(x)= \left(1 + \frac{1}{x}\right)^x\)
and trace it out for large values of
\(x\)
. The curve essentially flattens out as
\(x\)
increases.
[IMG]
Here is a graph of
\(g\)
along with a graph of the horizontal line given by
\(y = e\)
.
[IMG]
As the input grows, the two graphs are difficult to distinguish. The function
\(g\)
has a horizontal asymptote of
\(e\)
. (The exact value of the horizontal asymptote for
\(g\)
is one way to
define
the number
\(e\)
.)
Point out that, when
\(x\)
is a very large value,
\(\frac 1x\)
is very small (as we saw in function
\(d\)
). So in the expression
\(\left(1 + \frac{1}{x}\right)^x\)
, we are expressing
\(e\)
as
\(e \approx(1 + \text{tiny})^{\text{huge}}\)
. (It is critical that the "huge" number is the multiplicative inverse of the "small" number.)
The values of
\(h\)
and
\(k\)
are very close for very small, positive values of
\(x\)
. In other words, raising
\(e\)
to an increasingly small positive exponent
\(x\)
gives essentially the same values as adding 1 to
\(x\)
. Display a graph such as the following, or use graphing technology and zoom in toward smaller and smaller values of
\(x\)
.
[IMG]
Tell students that these behaviors turn out to be useful in mathematical and scientific applications, and they will learn a lot more about them in future math courses.
Conversing, Representing: MLR 7 Compare and Connect.
Invite students to make connections between the various representations and approaches. Ask students compare how the number
\(e\)
is represented in the table or graph of the function
\(g\)
. Listen for and amplify the language students use to describe how the value of
\(g\)
approaches the number
\(e\)
in the table or graph.
Design Principle(s): Maximize meta-awareness; Optimize output (for comparison)


correct_option_index: 6