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: Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
problem_activity: Activity
This activity prompts students to graph equations to solve problems and think about appropriate values of their variables. The context is the thickness and the area of a sheet of paper that is repeatedly folded.
For both equations, only whole numbers make sense as inputs. Students consider why this makes sense in this context. (This is in contrast, for example, to the price of producing solar energy in the earlier activity. While the data given is on an annual basis, it made sense to “connect the dots” because there is a price for producing solar energy at any given time.) Without instruction about how to create a discrete graph with their graphing technology, students are likely to produce a continuous graph. This is a good opportunity to discuss how to interpret a model for a particular context (MP4).
Look for the different ways students use to find out how many folds are needed to reach a certain thickness. Identify students who used a table, a graph, or an equation so they can share later.
Launch
Give each student a sheet of paper. Ask them to fold it in half, and in half again, as many times as they can. (To save paper, you might have students do this in pairs.) Once they have folded it as many times as they can, ask a few students to share how many times they folded it, and estimate the thickness of the folded paper. Tell students that they are going to investigate the relationship between the number of folds and the thickness of the paper.
Provide access to graphing technology. This activity requires students to be mindful about the scale or graphing window. The activities that specifically target that skill are optional, so this gives students a chance to practice scaling the axes.
Reading, Writing, Speaking: MLR3 Clarify, Critique, Correct.
Present an incorrect answer and explanation for the question “How many folds does it take before the folded sheet of paper is more than 1 mm thick?” Consider using this ambiguous response: “It takes 4.5 folds, because I looked on the graph to check the points.” Ask students to identify the error, critique the reasoning, and write an improved explanation. As students discuss with a partner, listen for students who identify and clarify the ambiguous language in the statement. Invite students to share their critiques and corrected explanations with the class. Listen for and amplify the language students use to describe the point using both the graph and a created table. This helps students evaluate, and improve on, the written mathematical arguments of others, as they clarify their understanding of discrete graphs.
Design Principle(s): Optimize output (for explanation); Maximize meta-awareness
Student Facing
The thickness
\(t\)
in millimeters of a folded sheet of paper after it is folded
\(n\)
times is given by the equation
\(t = (0.05) \boldcdot 2^n\)
.
What does the number
\(0.05\)
represent in the equation?
Use graphing technology to graph the equation
\(t = (0.05) \boldcdot 2^n\)
.
How many folds does it take before the folded sheet of paper is more than 1 mm thick? How many folds before it is more than 1 cm thick? Explain how you know.
The area of a sheet of paper is 93.5 square inches.
Find the remaining, visible area of the sheet of paper after it is folded in half once, twice, and three times.
Write an equation expressing the visible area
\(a\)
of the sheet of paper in terms of the number of times it has been folded
\(n\)
.
Use graphing technology to graph the equation.
In this context, can
\(n\)
take negative values? Explain your reasoning.
Can
\(a\)
take negative values? Explain your reasoning.
Student Response
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Student Facing
Are you ready for more?
Using the model in this task, how many folds would be needed to get 1 meter in thickness? 1 kilometer in thickness?
Do some research: what is the current world record for the number of times humans were able to fold a sheet of paper?
Student Response
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click here to register or sign in
for free access to Extension Student Response.
Anticipated Misconceptions
If students have trouble organizing their data points, suggest that they use a table.
Activity Synthesis
Focus the discussion on:
How students identified the growth factor, given an equation or a description.
How students determined how many folds are needed to reach a certain thickness. Invite previously identified students to share their strategies.
The meaningful values of the variables in the equations:
Highlight that a discrete graph is more appropriate in both situations: only the whole-number values make sense for the number of folds because it is not possible to fold the paper a fractional number of times.
Practically speaking, a large number of folds, such as 36, is not possible.
Demonstrate, using graphing technology, how to graph the equations and invite students to share how they chose an appropriate graphing window. If students just typed an equation, most likely the tool plotted a continuous curve. This is an opportunity to talk about interpreting the continuous graph in this context: really, only whole-number values for
\(n\)
are meaningful. Time permitting, you might demonstrate how to graph only whole number values of
\(n\)
using your graphing technology of choice. Typically, this is done by using the technology to generate a table based on a rule. Here is an example:
[IMG]
Time permitting, consider asking students if the equations
\(t = (0.05)\boldcdot 2^n \)
and
\(a = (93.5) \boldcdot \left(\frac{1}{2}\right)^n \)
could be plotted on the same set of axes. Even though the variable
\(n\)
(the number of folds) is the same for the two equations, the thickness
\(t\)
and the area
\(a\)
are mesaured in different units and so cannot be plotted on the same axes.


aligns: no