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.


Mathematical concept/skill options: 
A. Write and interpret numerical expressions.
B. Generate and analyze patterns.
C. Analyze patterns and relationships.
D. Use properties of operations to generate equivalent expressions.
E. Use the four operations with whole numbers to solve problems.
F. Work with radicals and integer exponents.
G. Solve systems of equations.
H. Represent and solve problems involving multiplication and division.
I. Perform arithmetic operations on polynomials.
J. Understand the relationship between zeros and factors of polynomials.
K. Understand properties of multiplication and the relationship between multiplication and division.
L. Solve real-life and mathematical problems using numerical and algebraic expressions and equations.
M. Solve problems involving the four operations, and identify and explain patterns in arithmetic.
N. Use polynomial identities to solve problems.
O. Apply and extend previous understandings of arithmetic to algebraic expressions.
P. Rewrite rational expressions.
Q. Interpret the structure of expressions.
R. Add and subtract within 20.
S. Analyze and solve linear equations and pairs of simultaneous linear equations.
T. Represent and analyze quantitative relationships between dependent and independent variables.
U. Solve equations and inequalities in one variable.
V. Create equations that describe numbers or relationships
W. Reason about and solve one-variable equations and inequalities.
X. Represent and solve equations and inequalities graphically.
Y. Understand solving equations as a process of reasoning and explain the reasoning.
Z. Understand and apply properties of operations and the relationship between addition and subtraction.
AA. Gain familiarity with factors and multiples.
AB. Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.
AC. Work with addition and subtraction equations.
AD. Multiply and divide within 100.
AE. Work with equal groups of objects to gain foundations for multiplication.
AF. Represent and solve problems involving addition and subtraction.
AG. Write expressions in equivalent forms to solve problems.
AH. Understand the connections between proportional relationships, lines, and linear equations.


Thought: This task teaches students to understand addition as putting together counters, or adding to already placed counters.
Answer: AB

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


Mathematical concept/skill options: 
A. Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.
B. Reason quantitatively and use units to solve problems.
C. Apply and extend previous understandings of numbers to the system of rational numbers.
D. Represent complex numbers and their operations on the complex plane.
E. Perform arithmetic operations with complex numbers.
F. Compute fluently with multi-digit numbers and find common factors and multiples.
G. Represent and model with vector quantities.
H. Perform operations on vectors.
I. Use properties of rational and irrational numbers.
J. Use complex numbers in polynomial identities and equations.
K. Extend the properties of exponents to rational exponents.
L. Know that there are numbers that are not rational, and approximate them by rational numbers.
M. Perform operations on matrices and use matrices in applications.
N. Apply and extend previous understandings of multiplication and division to divide fractions by fractions.


Thought: This problem involves understanding fractions, multiplication, and division.
Answer: A

Example 3:
Problem/activity:
Find the general solution for:
$${\mathrm{cos}\left({\pi\over2}+x\right)=\mathrm{tan}{\pi\over4}}$$


Mathematical concept/skill options: 
A. Prove and apply trigonometric identities.
B. Model periodic phenomena with trigonometric functions.
C. Interpret functions that arise in applications in terms of the context.
D. Build a function that models a relationship between two quantities.
E. Define, evaluate, and compare functions.
F. Use functions to model relationships between quantities.
G. Understand the concept of a function and use function notation.
H. Analyze functions using different representations.
I. Extend the domain of trigonometric functions using the unit circle.
J. Interpret expressions for functions in terms of the situation they model.
K. Build new functions from existing functions.
L. Construct and compare linear and exponential models and solve problems.


Thought: This problem applies trigonometric identities, including the angle addition formula for cosine.
Answer: A

Now, assign the following problem or activity to one or more relevant mathematical concepts/skills it teaches.
shot_info: im_task_001477

K

fl_problem_001097

7

fl_problem_001894

HS_0
options: Work with equal groups of objects to gain foundations for multiplication.

Add and subtract within 20.

Work with addition and subtraction equations.

Reason about and solve one-variable equations and inequalities.

Generate and analyze patterns.

Understand solving equations as a process of reasoning and explain the reasoning.

Solve problems involving the four operations, and identify and explain patterns in arithmetic.

Represent and solve equations and inequalities graphically.

Create equations that describe numbers or relationships

Use polynomial identities to solve problems.

Interpret the structure of expressions.

Represent and solve problems involving multiplication and division.

Write expressions in equivalent forms to solve problems.

Gain familiarity with factors and multiples.

Understand the relationship between zeros and factors of polynomials.

Write and interpret numerical expressions.

Represent and solve problems involving addition and subtraction.

Understand properties of multiplication and the relationship between multiplication and division.

Work with radicals and integer exponents.

Rewrite rational expressions.

Represent and analyze quantitative relationships between dependent and independent variables.

Solve real-life and mathematical problems using numerical and algebraic expressions and equations.

Understand and apply properties of operations and the relationship between addition and subtraction.

Solve systems of equations.

Use the four operations with whole numbers to solve problems.

Understand the connections between proportional relationships, lines, and linear equations.

Use properties of operations to generate equivalent expressions.

Apply and extend previous understandings of arithmetic to algebraic expressions.

Analyze patterns and relationships.

Analyze and solve linear equations and pairs of simultaneous linear equations.

Perform arithmetic operations on polynomials.

Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.

Solve equations and inequalities in one variable.

Multiply and divide within 100.
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
Teachers with a valid work email address can
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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.


correct_option_index: 10

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