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: Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ³ 1.

Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).

Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
problem_activity: Activity
This activity serves as a brief and light introduction to how
\(e\)
is visible in the behaviors of certain functions. The standards addressed in this course don’t require a deep understanding of the meaning of
\(e\)
, so the observations from this activity should not be assessed.
\(e\)
has a special connection to the expression
\(\left(1+\frac{1}{x}\right)^x\)
. Jumping to this expression would likely be an opaque process for most students, so they first investigate the range of behavior of its component pieces, for very small and very large positive values of
\(x\)
.
Entering increasingly larger or smaller values of
\(x\)
in the functions and making observations about them is a chance to look for and express regularity through repeated reasoning (MP8).
Launch
Tell students that they are to analyze the behavior of some functions and see what observations could be made about
\(e\)
along the way. Provide access to graphing technology and spreadsheet tools.
Arrange students in groups of 2. Tell students that they each should analyse the functions in different ways and then compare results. For example, one student could examine the functions numerically, using a table or a spreadsheet, while another student could examine them graphically.
Consider naming some tiny positive values of
\(x\)
(for example, 0.1, 0.01, and 0.001) and large values of
\(x\)
(for example, 100, 1,000, and 100,000) to students who may benefit from concrete examples.
Action and Expression: Internalize Executive Functions.
Provide students with a graphic organizer to capture their observations. Provide seven columns for students to organize their work. The first column label should be “function,” while the remaining six columns are divided into two sets; three for the three “tiny input” numbers and three for the “large input numbers.” Ask students to select three tiny input numbers and three large input numbers (either from the numbers given, or their own). Demonstrate sequencing the numbers for the column headers from left to right getting tinier, and from left to right getting larger. Explain to students that some may not need all six columns to draw conclusions. Students may especially want to capture observations and calculations for all six input values for
\(f(x)\)
and
\(g(x)\)
.
Supports accessibility for: Language; Organization
Student Facing
Here are some functions. For each function, describe, in words, the outputs for very tiny, positive values of
\(x\)
and for very large values of
\(x\)
.
\(a(x) = 1^x\)
\(b(x) = \text-x\)
\(d(x) = \frac{1}{x}\)
\(f(x) = \left(\frac{1}{x}\right)^x\)
\(g(x) = \left(1 + \frac{1}{x}\right)^x\)
\(h(x) = e^{x}\)
\(k(x) = 1 + x\)
Remember that
\(e \approx 2.718\)
. What does the function
\(g\)
have to do with the number
\(e\)
?
What do you notice about the relationship between
\(h\)
and
\(k\)
for very small, positive values of
\(x\)
?
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?
Complete the table to show the value of each expression to the nearest hundred-thousandth. Two entries have already been completed as an example.
| \(x\)  | \(2^x\) | \(e^x\) |        \(3^x\)        |
|--------|---------|---------|-----------------------|
|  0.1   | 1.07177 | 1.10517 | \(\phantom{1.11612}\) |
|  0.01  |         |         |                       |
| 0.001  |         |         |                       |
| 0.0001 |         |         |                       |
What do you notice about the 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 Extension Student Response.
Anticipated Misconceptions
Students may substitute
\(0\)
for
\(x\)
in order to see what happens for small positive values of
\(x\)
. Focus their attention on functions
\(d\)
,
\(f\)
, and
\(g\)
where the
\(0\)
does not belong to the domain. For the other functions, encourage students to look at a graph (using technology for example). Does the value when
\(x = 0\)
help figure out what values each function takes for small positive values of
\(x\)
?
Activity Synthesis
Invite students to share their observations for the first question and the supporting work (a table, a spreadsheet, or graphs) that led to these observations, if available.
Focus the conversation on the responses to the last two questions. If not already observed by students, highlight that:
The value of function
\(g\)
gets closer and closer to
\(e\)
as
\(x\)
gets larger and larger. Display a graph representing
\(g(x)= \left(1 + \frac{1}{x}\right)^x\)
and trace it out for large values of
\(x\)
. The curve essentially flattens out as
\(x\)
increases.
[IMG]
Here is a graph of
\(g\)
along with a graph of the horizontal line given by
\(y = e\)
.
[IMG]
As the input grows, the two graphs are difficult to distinguish. The function
\(g\)
has a horizontal asymptote of
\(e\)
. (The exact value of the horizontal asymptote for
\(g\)
is one way to
define
the number
\(e\)
.)
Point out that, when
\(x\)
is a very large value,
\(\frac 1x\)
is very small (as we saw in function
\(d\)
). So in the expression
\(\left(1 + \frac{1}{x}\right)^x\)
, we are expressing
\(e\)
as
\(e \approx(1 + \text{tiny})^{\text{huge}}\)
. (It is critical that the "huge" number is the multiplicative inverse of the "small" number.)
The values of
\(h\)
and
\(k\)
are very close for very small, positive values of
\(x\)
. In other words, raising
\(e\)
to an increasingly small positive exponent
\(x\)
gives essentially the same values as adding 1 to
\(x\)
. Display a graph such as the following, or use graphing technology and zoom in toward smaller and smaller values of
\(x\)
.
[IMG]
Tell students that these behaviors turn out to be useful in mathematical and scientific applications, and they will learn a lot more about them in future math courses.
Conversing, Representing: MLR 7 Compare and Connect.
Invite students to make connections between the various representations and approaches. Ask students compare how the number
\(e\)
is represented in the table or graph of the function
\(g\)
. Listen for and amplify the language students use to describe how the value of
\(g\)
approaches the number
\(e\)
in the table or graph.
Design Principle(s): Maximize meta-awareness; Optimize output (for comparison)


correct_option_index: 2