few_shots: 
Example 1:
Concept/skill: 
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.

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?


Answer: 
Yes. 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.

Example 2:
Concept/skill: 
Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.

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
$$-\frac{1}{4}$$


Answer: 
Yes. 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).

Example 3:
Concept/skill: 
Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.

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?


Answer: 
No. Completing neither of these problems requires replacing one equation by the sum of that equation and a multiple of the other equation.

Now, determine whether the following problem or activity can enable students to understand the concepts or skills in the provided description.

shot_info: fl_problem_000972_pos_0

6

fl_problem_001097_pos_0

7

fl_problem_001392_neg_0

8
standard_description: For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.
problem_activity: In Class Launch
Use after Unit 4, Lesson 16
Ask students to give examples of situations where there is exponential change (population growth, compounding interest, radioactive decay). Write down students’ ideas for all to see.
Then ask students for an example of an exponential function (
\(y=2e^{\text-t}\)
). Write it for all to see. Here are some possible questions for discussion:
“What situation could this function model?” (Since the exponent is negative, this models decay. The coefficient of the exponential part is 2, so the initial value is 2. The amount is divided by
\(e\)
every time
\(t\)
increases by 1. So it’s a small amount of something and it’s decreasing. How fast it decreases depends on what
\(t\)
represents.)
“What questions about a situation could this function help you answer?” (I could find out what
\(t\)
is when the value of the function is a specific number, like 0.01. If
\(t\)
represents time, I could find out how often the value of the function doubles.)
Tell students that in this task they will ask and answer their own questions about a situation that can be modeled with an exponential function.
Teacher Instructions
After students have finished the task, they should see what questions were asked about each situation. One way to do this would be to have groups that looked at the same situation get together to compare their questions and the assumptions they made about the situation. Groups that made up their own situations can get together and share what they did. Then one or two representatives from each large group can summarize the discussion for the whole class.
Student-Facing Statement
Choose one of these situations to think about, or write a similar one of your own:
There are currently 61 million cars in a certain country, increasing by 6% annually.
A vampire enters a town. He looks around for about an hour, and then chooses someone to bite. The person he bit is now also a vampire. Both of them take an hour to choose new victims, and then they each bite someone. All the vampires continue choosing victims in the same way.
The population in millions,
\(W\)
, of the country of West Mathlandia is modeled by
\(W(x) = 17.9e^{0.0015x}\)
, where
\(x\)
is the number of years since September 1998. The population in millions,
\(E\)
, of East Mathlandia is modeled by
\(E(x) = 13.5e^{0.0128x}\)
.
There are 10,000 bacteria in a petri dish. The population of bacteria declines to 4,000 in 4 hours.
An artifact is discovered. It has 57% of the carbon-14 it originally contained. Carbon-14 decays at the rate of 0.0125% per year.
The number of books,
\(B\)
, in a small library increases according to the function
\(B = 5,\!300e^{0.02t}\)
, where
\(t\)
is measured in years.
The growth in the population,
\(P\)
, of a certain species of rodent at a landfill is modeled by the exponential function
\(P(t)= 648e^{0.016t}\)
, where
\(t\)
is the number of years since 1987.
The initial mass of a certain compound is 448 g, and every 27 hours it decreases by half.
Write down questions you could ask about the situation you chose. Write as many questions as you can think of. As a group, you will choose some interesting questions to answer. If you would need to do calculations, make a sketch, or draw a graph in order to answer a question, then it’s probably an interesting question.
You should answer 2 questions for every person in your group. Multiple people can work on a question together—each person in the group does not have to answer 2 questions by themselves.
Write a report to explain your questions and answers. The report should include:
A context for your situation. Write a story that includes all the relevant information. If the description of the situation is missing some details, fill them in however you want.
An equation to describe the situation, with an explanation of what each part of the equation represents.
A labeled graph of the situation.
The questions you chose and your answers. Include your process for finding the answers. Say what the answers mean in the context of this situation.
Teacher Instructions
After students have finished the task, they should see what questions were asked about each situation. One way to do this would be to have groups that looked at the same situation get together to compare their questions and the assumptions they made about the situation. Groups that made up their own situations can get together and share what they did. Then one or two representatives from each large group can summarize the discussion for the whole class.
Student-Facing Statement
Choose one of these situations to think about, or write a similar one of your own:
A vampire enters a town. He looks around for about an hour, and then chooses someone to bite. The person he bit is now also a vampire. Both of them take an hour to choose new victims, and then they each bite someone. All the vampires continue choosing victims in the same way.
The population in millions,
\(W\)
, of the country of West Mathlandia is modeled by
\(W(x) = 17.9e^{0.0015x}\)
, where
\(x\)
is the number of years since September 1998. The population in millions,
\(E\)
, of East Mathlandia is modeled by
\(E(x) = 13.5e^{0.0128x}\)
.
There are 10,000 bacteria in a petri dish. The population of bacteria declines to 4,000 in 4 hours.
The number of books,
\(B\)
, in a small library increases according to the function
\(B = 5,\!300e^{0.02t}\)
, where
\(t\)
is measured in years.
The growth in the population,
\(P\)
, of a certain species of rodent at a landfill is modeled by the exponential function
\(P(t)= 648e^{0.016t}\)
, where
\(t\)
is the number of years since 1987.
Write down questions you could ask about the situation you chose. Write as many questions as you can think of. As a group, you will choose some interesting questions to answer. If you would need to do calculations, make a sketch, or draw a graph in order to answer a question, then it's probably an interesting question. To get you started, here are some example questions you could ask about each situation:
If there are 200 people in the town, how long will it be before all of them are vampires?
Will the populations of East and West Mathlandia ever be the same, and if so, when?
How much does the population of bacteria decline each hour?
How long will it be before the library has 1 million books?
How many rodents were born between 1987 and now?
You should answer 2 questions for every person in your group. Multiple people can work on a question together—each person in the group does not have to answer 2 questions by themselves.
Write a report to explain your questions and answers. The report should include:
A context for your situation. Write a story that includes all the relevant information. If the description of the situation is missing some details, fill them in however you want.
An equation to describe the situation, with an explanation of what each part of the equation represents.
A labeled graph of the situation.
The questions you chose and your answers. Include your process for finding the answers. Say what the answers mean in the context of this situation.


aligns: yes