few_shots: 
Example 1:
Concept/skill: 
Recognize and represent proportional relationships between quantities.

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: 
No. Usually problems that teach recognizing and representing proportional relationships between quantities involve deciding whether two quantities are in a proportional relationship or identifying the constant of proportionality, and then representing proportional relationships by equations or explaining proportional relationship using coordinate graphs.

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_neg_0

6

fl_problem_001097_pos_0

7

fl_problem_001392_neg_0

8
standard_description: Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.
problem_activity: Activity
This activity prompts students to again compare a linear function with an exponential one, but this time without a context, and the exponential function grows much more slowly over a long period of time. Even if students predict that the exponential function will grow more quickly because it is exponential, they still need to decide which one reaches 2,000 first. To do that, students may try:
continuing the table of values with increasingly larger values
using graphing technology
finding when
\(f(x) = 2,\!000\)
and substituting this value (
\(x=1,\!000\)
) into
\(g(x)\)
Making graphing and spreadsheet technology available gives students an opportunity to choose appropriate tools strategically (MP5).
Launch
Present the two equations that define two functions
\(f\)
and
\(g\)
:
\(f(x) = 2x\)
and
\(g(x) = (1.01)^x\)
. Give students a moment to share what they notice and wonder about the functions. Ask questions such as:
What is happening in the first function? What about in the second function?
Which function do you think will reach a value of 2,000 first?
Representing, Conversing: MLR7 Compare and Connect.
Ask students to prepare a visual display that shows their mathematical thinking in response to the question “Which function do you think will reach a value of 2,000 first?” Invite previously identified students to share their representations. In groups of 2, ask students to compare their strategies and discuss how the strategies are the same or different. Consider using the prompt “Which strategy do you believe is the most efficient in determining a solution?” Listen for and amplify observations of advantages and disadvantages to different approaches, such as using a particular approach for a certain time interval. This will help students connect different approaches and determine the effectiveness of each one through partner and whole-class discussion.
Design Principle(s): Cultivate conversation; Maximize meta-awareness
Action and Expression: Provide Access for Physical Action.
Provide access to tools and assistive technologies, such as a graphing calculator or graphing software. Some students may benefit from a checklist or list of steps to be able to use the calculator or software.
Supports accessibility for: Organization; Conceptual processing; Attention
Student Facing
Complete the table of values for the functions
\(f\)
and
\(g\)
.
| \(x\) | \(f(x)\) | \(g(x)\) |
|-------|----------|----------|
|   1   |          |          |
|  10   |          |          |
|  50   |          |          |
|  100  |          |          |
|  500  |          |          |
|       |          |          |
|       |          |          |
|       |          |          |
Based on the table of values, which function do you think grows faster? Explain your reasoning.
Which function do you think will reach a value of 2,000 first? Show your reasoning. If you get stuck, consider increasing
\(x\)
by 100 a few times and record the function 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 Student Response.
Student Facing
Are you ready for more?
Consider the functions
\(g(x)=x^5\)
and
\(f(x)=5^x\)
. While it is true that
\(f(7)>g(7)\)
, for example, it is hard to check this using mental math. Find a value of
\(x\)
for which properties of exponents allow you to conclude that
\(f(x)>g(x)\)
without a calculator.
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
Select students who used contrasting approaches to share. The exponential function
\(g\)
grows sufficiently slowly that it looks like it will not catch up with or ever overtake the linear function
\(f\)
. This is true for a table of values as well as using a graph. Invite students who used contrasting methods to share, in the order shown in the Activity Narrative.
Emphasize that the table needs to be continued for a long time to identify when the values of
\(g\)
become greater than those of
\(f\)
. Similarly, the window of the graph needs to be chosen carefully. A very efficient method is to identify that
\(f(1,\!000)=2,\!000\)
and then evaluate
\(g(1,\!000)\)
. Since
\(g(1,\!000)\)
is much greater than 2,000,
\(g\)
reaches 2,000 first.
If not already illustrated by students who used graphing, show this dynamic sketch and zoom out:
Though for quite a while it doesn't seem like the values of
\(g\)
would ever catch up with those of
\(f\)
, if the growth is allowed to take place long enough, exponential eventually surpasses linear.


aligns: yes