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:
Task
 Arrange the fractions in order from least to greatest. Explain your answer with a picture.
 $\frac{1}{5}, \frac{1}{7}, \frac{1}{3} $
 $\frac{2}{5}, \frac{2}{7}, \frac{2}{3} $
 $\frac{5}{6}, \frac{3}{6}, \frac{1}{6} $
 $\frac{5}{12}, \frac{8}{12}, \frac{4}{12} $

Mathematical concept/skill options: 
A. Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.
Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.
B. Extend understanding of fraction equivalence and ordering.
Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.
C. Use equivalent fractions as a strategy to add and subtract fractions.
D. Apply and extend previous understandings of multiplication and division to multiply and divide fractions.
E. Understand decimal notation for fractions, and compare decimal fractions.
Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.
F. Develop Understanding of fractions as numbers.
Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8.


Thought: This problem involves understanding fractions as numbers by comparing their quantities.
Answer: F

Example 3:
Problem/activity:
Problem 1
Solve using the standard algorithm. If you get stuck, use an area model and/or the partial products algorithm to help. Then assess the reasonableness of your answer.
a.
$$5,803\times46$$
b.
$$63\times2,116$$
Problem 2
Dunkin' introduced a croissant doughnut in 2014. In the first three months of its debut, they sold 7,870 dozen of them per day. How many croissant doughnuts were sold in the first three months?


Mathematical concept/skill options: 
A. Use place value understanding and properties of operations to perform multi-digit arithmetic. (A range of algorithms may be used.)
B. Understand place value.
C. Understand the place value system.
D. Extend the counting sequence.
E. Work with numbers 11–19 to gain foundations for place value.
F. Use place value understanding and properties of operations to perform multi-digit arithmetic.
Grade 4 expectations in this domain are limited to whole numbers less than or equal to 1,000,000.
G. Perform operations with multi-digit whole numbers and with decimals to hundredths.
H. Generalize place value understanding for multi-digit whole numbers.
Grade 4 expectations in this domain are limited to whole numbers less than or equal to 1,000,000.
I. Use place value understanding and properties of operations to add and subtract.


Thought: This set of problems asks students to perform multiplication, which is an operation, on multi-digit whole numbers.
Answer: G

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

K

im_task_000157

3

fl_problem_000674

5
options: Interpret expressions for functions in terms of the situation they model.

Build a function that models a relationship between two quantities.

Model periodic phenomena with trigonometric functions.

Interpret functions that arise in applications in terms of the context.

Understand the concept of a function and use function notation.

Build new functions from existing functions.

Construct and compare linear and exponential models and solve problems.

Extend the domain of trigonometric functions using the unit circle.

Analyze functions using different representations.

Prove and apply trigonometric identities.

Define, evaluate, and compare functions.

Use functions to model relationships between quantities.
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: 4