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: Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.
problem_activity: Activity
This activity is the first time students study a rational function with a non-zero horizontal asymptote. The first part of the activity is for students to make sense of the context while reasoning about input-output pairs. The questions in the activity and the following discussion ask students to reason about why the cost per book trends toward
$
4 from first a context perspective, and then by rewriting the expression for the function and interpreting it within the context.
Monitor for students who have clear explanations regarding what the end behavior says about the context.
Launch
Display the opening statement and graph for all to see. Give students 1 minute to read the statement and then ask, “In the expression for
\(c(x)\)
, there is division by
\(x\)
since it gives the cost per book. What do you think the
\(120+4x\)
means?” (The
\(120+4x\)
is the cost to make
\(x\)
books, which is
$
4 per book and a
$
120 set-up fee.) Once students understand the different parts of the expression, allow them to continue with the first question in the activity.
If time allows, encourage students to calculate more accurate responses to the first three questions by using the equation of the function. Otherwise, approximating using the graph is the expectation.
Conversing, Writing: MLR2 Collect and Display.
Before students begin writing a response to the last question, invite them to discuss their thinking with a partner. Listen for and collect vocabulary, gestures, and phrases students use to describe the function’s behavior as it approaches infinity. Capture student language that reflects a variety of ways to describe the behavior of the function near the non-zero horizontal asymptote. Write the students’ words on a visual display or add it to the display from the previous lesson and update it throughout the remainder of the lesson, with phrases such as “gets close to 4,” “never reaches,” and “almost flat.” Add the term “horizontal asymptote” to the chart when introduced in the activity synthesis. Remind students to borrow language from the display as needed. This will help students read and use mathematical language during their partner and whole-group discussions.
Design Principle(s): Support sense-making
Representation: Internalize Comprehension.
Use color coding and annotations to highlight connections between representations in a problem. When students have seen that
\(\frac{4x+120}{x}\)
can also be expressed as
\(4+\frac{120}{x}\)
, they can sketch the graph, record both expressions on a display nearby, annotate to show the horizontal asymptote and end behavior, and record the terms in the expressions associated with each. Students can use this as a reference in the next activity.
Supports accessibility for: Organization; Conceptual processing; Memory
Student Facing
Let
\(c\)
be the function that gives the average cost per book
\(c(x)\)
, in dollars, when using an online store to print
\(x\)
copies of a self-published paperback book. Here is a graph of
\(c(x)= \tfrac{120+4x}{x}.\)
[IMG]
What is the approximate cost per book when 50 books are printed? 100 books?
The author plans to charge
$
8 per book. About how many should be printed to make a profit?
What is the value of
\(c(x)\)
when
\(x=\frac{1}{2}\)
? How does this relate to the context?
What does the end behavior of the function say about the context?
Student Response
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for free access to Student Response.
Anticipated Misconceptions
Students who are only looking at the graph might think that the end behavior is close to
$
0. Suggest substituting a large number for
\(x\)
or looking at a table to see the value of
\(c(x)\)
for large values of
\(x\)
.
Activity Synthesis
The purpose of this discussion is for students to learn about and discuss what a horizontal asymptote is, building on what students already know about the end behavior of polynomial functions.
Begin the discussion by selecting students to share how they worked out solutions to the first 3 questions. If any students used the equation to answer these questions exactly and time allows, invite them to share their process, particularly for the question asking how many books must be printed in order to make a profit. Students will have more opportunities to solve rational equations algebraically in future lessons, so this does not need to be a focus at this time.
Select previously identified students to share what the end behavior of the function says about the context. Then, ask students how we could rewrite
\(\frac{120+4x}{x}\)
to make the end behavior more obvious when looking at the expression for
\(c\)
. After quiet work time, invite students to share their ideas, recording these for all to see. Hopefully, at least one student has written
\(4+\frac{120}{x}\)
, but if not, help guide students to this form by suggesting decomposing the single fraction into two fractions.
Once students understand how the new form was written, ask, “Which term in
\(4+\frac{120}{x}\)
uncovers the end behavior?” (The 4 uncovers the end behavior of
\(c\)
because when
\(x\)
gets larger and larger in the positive direction,
\(\frac{120}{x}\)
gets closer to 0.) Tell students that this type of end behavior is due to a
horizontal asymptote
. For large numbers of books printed, the value of
\(c(x)\)
will get closer and closer to
$
4 but never quite reach it due to the
$
120 set-up fee that is divided among the total number of books printed.
Conclude the discussion by making sure students understand that not all rational functions have horizontal asymptotes (just think back to the graphs of the cylinder), but those that do can be written a certain way, which is the focus of the next activity.


aligns: no