few_shots: Example 1:
Problem/activity:
You’re mixing ingredients for cookies. The recipe says to combine 6 tablespoons, or ¾ stick, of butter with 1 cup of sugar. You accidentally mix in a whole stick of butter (8 tablespoons) with the cup of sugar. How can you fix this?


Standard options: 
A. Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, “The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.” “For every vote candidate A received, candidate C received nearly three votes.”
B. Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.
C. Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠0, and use rate language in the context of a ratio relationship. For example, “This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar.” “We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.” (Expectations for unit rates in this grade are limited to non-complex fractions.)


Your response:
Thought: This problem solves a real-world problem involving mixing ingredients for cookies. A student doing this problem would need to reason about equivalent ratios of butter amounts to sugar amounts.
Answer: B

Example 2:
Problem/activity:
For each pair of numbers, determine if the values are equivalent. Explain your reasoning for each one.
a.
$$-0.25$$
and
$$\frac{1}{-4}$$
b.
$$-\left ( \frac{1}{4} \right )$$
and
$$\frac{-1}{-4}$$
c.
$$\frac{-1}{4}$$
and
$$-\frac{1}{4}$$


Standard options: 
A. Solve real-world and mathematical problems involving the four operations with rational numbers. (Computations with rational numbers extend the rules for manipulating fractions to complex fractions.)
B. Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.
C. Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.


Your response:
Thought: This problem involves understanding fractions and division to divide rational numbers, such as 1, -4, 4, and -1. This problem also involves an understanding of multiplication: if p and q are integers, then -(p/q) = (-p)/q = p/(-q).
Answer: B

Example 3:
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?


Standard options: 
A. Solve linear equations in one variable.
B. Analyze and solve pairs of simultaneous linear equations.


Your response:
Thought: The last three rows of the table in the first problem involves solving pairs of simultaneous linear equations. The second word problem can also be represented as two simultaneous linear equations.
Answer: B

Now, assign the following problem or activity to one or more relevant standards it aligns with.
shot_info: fl_problem_000972

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fl_problem_001097

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fl_problem_001392

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options: Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.

Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).

Distinguish between situations that can be modeled with linear functions and with exponential functions.

For exponential models, express as a logarithm the solution to ab^(ct) = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.
problem_activity: Optional activity
This activity is optional. Use this activity to allow students to practice analyzing graphs of exponential functions involving
\(e\)
and setting an appropriate graphing window on their graphing technology in order to produce useful graphs.
Students have graphed exponential functions and analyzed the graphs to solve problems prior to this point. Though the functions here use base
\(e\)
, the analytical work is fundamentally the same as that when the base is another number. Students will also have opportunities to graph exponential functions with base
\(e\)
later in the unit.
Launch
Provide access to devices that can run Desmos or other graphing technology.
Writing, Listening, Conversing: MLR1 Stronger and Clearer Each Time.
Use this routine to help students improve their written responses to the first question. Give students time to meet with 2–3 partners to share and receive feedback on their responses. Display feedback prompts that will help students strengthen their ideas and clarify their language. For example, “How did you get . . . ?”, “Why did you . . . ?”, and “How do you know . . . ?” Invite students to go back and revise or refine their written responses based on the feedback from peers. This will help students justify their reasoning for the population of bacteria 10 hours after the scientist put the initial bacteria in the dish.
Design Principle(s): Optimize output (for justification); Cultivate conversation
Student Facing
Use graphing technology to graph the function defined by
\(f(t) = 50 \boldcdot e^{(0.25t)}\)
. Adjust the graphing window as needed to answer these questions:
The function
\(f\)
models the population of bacteria in
\(t\)
hours after it was initially measured. About how many bacteria were in the dish 10 hours after the scientist put the initial 50 bacteria in the dish?
About how many hours did it take for the number of bacteria in the dish to double? Explain or show your reasoning.
Use graphing technology to graph the function defined by
\(p(t)=192 \boldcdot e^{(0.014t)}\)
. Adjust the graphing window as needed to answer these questions:
The equation models the population, in millions, in the U.S.
\(t\)
years after 1964. What does the model predict for the population of the U.S. in 1974?
In which year does the model predict the population will reach 300 million?
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?
Research what the population of the U.S. was in the year the model predicted 300 million people. How far off was the model? What factors do you think account for the actual population in that year being different from the prediction of the model? In what year did the U.S. actually reach 300 million people?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Activity Synthesis
Focus the discussion on how students set the graphing window and how they used the graphs to answer the questions.
Depending on the graphing technology used, students may use the following strategies:
Visually estimate the value of the output given a certain input, or the input given an output.
Use a tracing tool to trace the graph and identify the coordinates when the input or output has a certain value.
To find an unknown input: graph a horizontal line at a given
\(y\)
-value (for example,
\(y=100\)
, a doubled bacteria population), find the intersection of that line and the graph of the exponential function, and identify the
\(x\)
-value at that intersection.
To find an unknown output: use a table or type something like
\(b(10)\)
after
\(b\)
was defined.
Some students may also notice that they could calculate the output value given an exponential equation and an input value, but had to rely on the graph to find an unknown input value because they were unsure how to solve for
\(x\)
when the equation has
\(e\)
as a base. Tell students that in a future lesson they will learn how to use logarithms to solve this type of equation algebraically similar to how they have used logarithms previously.


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