few_shots: Example 1:
Problem/activity:
Task
Materials
One 10-frame per pair of students (see example below)
[IMG]
Counters, 10 per pair of students
Decks of number cards 0-5 (with 4 of each number) or two 0-5 dice
Setup
This activity is done in pairs. The students sit side by side with 10-frame and deck of cards or dice. If they have cards, the cards are shuffled and placed face down in front of them.
Action
Player 1 flips a card or rolls the die. They place that many counters on the 10-frame. Player 2 flips a card or rolls the die. They place that many more counters on the 10-frame next to the counters already on the ten frame. The students count the total together.
A recording sheet can be added when the students have been introduced to equations and number sentences. For those students who can handle higher numbers, a deck of 0-10 cards and two 10-frames can be used.
This game can also be played alone.


Standard options: 
A. Represent addition and subtraction with objects, fingers, mental images, drawings (Drawings need not show details, but should show the mathematics in the problem. (This applies wherever drawings are mentioned in the Standards.)), sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations.
B. For any number from 1 to 9, find the number that makes 10 when added to the given number, e.g., by using objects or drawings, and record the answer with a drawing or equation.
C. Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or drawings, and record each decomposition by a drawing or equation (e.g., 5 = 2 + 3 and 5 = 4 + 1).
D. Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.
E. Fluently add and subtract within 5.


Thought: This task involves representing addition with objects such as number cards and dice.
Answer: A

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.


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:
Find the general solution for:
$${\mathrm{cos}\left({\pi\over2}+x\right)=\mathrm{tan}{\pi\over4}}$$


Standard options: 
A. Prove the Pythagorean identity sin^(2)(θ) + cos^(2)(θ) = 1 and use it to find sin(θ), cos(θ),or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.
B. Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.


Thought: An initial step when solving for x in this problem requires the use of the addition formula for cosine on the left side of the equation. Solving this problem does not require use of the Pythagorean identity.
Answer: B

Now, assign the following problem or activity to one or more relevant standards it aligns with.
shot_info: im_task_001477

K

fl_problem_001097

7

fl_problem_001894

HS_0
options: Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.

Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions.
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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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.


correct_option_index: 0