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: Understand the concept of a function and use function notation.

Extend the domain of trigonometric functions using the unit circle.

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

Build a function that models a relationship between two quantities.

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

Define, evaluate, and compare functions.

Analyze functions using different representations.

Use functions to model relationships between quantities.

Construct and compare linear and exponential models and solve problems.

Build new functions from existing functions.

Model periodic phenomena with trigonometric functions.

Prove and apply trigonometric identities.
problem_activity: Activity
Continuing the thinking from the warm-up, students now compare three functions whose graphs are vertical translations from one another and generalize their observations of the graphs and data tables into equations where one function is defined in terms of another.
Monitor for students making connections between representations as they write an equation for
\(g(x)\)
in terms of
\(h(x)\)
. For example, students may:
compare the graphs of
\(g\)
and
\(h\)
, perhaps sketching them on the same axes and then measuring the vertical distance between them
calculate the difference between the outputs from the table
combine the equation they wrote for
\(g(x)\)
in terms of
\(f(x)\)
and
\(h(x)=f(x)-2.5\)
to rewrite
\(g(x)\)
in terms of
\(h(x)\)
Launch
Representation: Internalize Comprehension.
Chunk this task into more manageable parts. Check in with students after the first 2–3 minutes of work time to look for students who notice the constant difference between the outputs of
\(f(x)\)
and
\(g(x)\)
. Check to make sure that students are filling out the
\(g(x)\)
column of the table correctly.
Supports accessibility for: Conceptual processing; Organization
Student Facing
The graph of function
\(g\)
is a vertical translation of the graph of polynomial
\(f\)
.
[IMG]
[IMG]
Complete the
\(g(x)\)
column of the table.
If
\(f(0) = \text-0.86\)
, what is
\(g(0)\)
? Explain how you know.
Write an equation for
\(g(x)\)
in terms of
\(f(x)\)
for any input
\(x\)
.
The function
\(h\)
can be written in terms of
\(f\)
as
\(h(x)=f(x)-2.5\)
. Complete the
\(h(x)\)
column of the table.
| \(x\) | \(f(x)\) | \(g(x)\) | \(h(x)=f(x)-2.5\) |
|-------|----------|----------|-------------------|
|  -4   |    0     |          |                   |
|  -3   |   -5.8   |          |                   |
| -0.7  |    0     |          |                   |
|  1.2  |   -3.3   |          |                   |
|   2   |    0     |          |                   |
Sketch the graph of function
\(h\)
.
[IMG]
Write an equation for
\(g(x)\)
in terms of
\(h(x)\)
for any input
\(x\)
.
Student Response
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Activity Synthesis
A key takeaway from this discussion is that students understand that a vertical translation from one graph to another corresponds to constant differences between the outputs of the functions for the same input. We can then generalize these point by point comparisons using function notation, where adding or subtracting from the output of the function resulting in a vertical translation of its graph.
Select previously identified students to share the connections they made between the representations in the order listed in the Activity Narrative. If time allows and no students used the equations for
\(g(x)\)
and
\(h(x)\)
in terms of
\(f(x)\)
to write a new equation for
\(g(x)\)
in terms of
\(h(x)\)
, invite them to do so now.


correct_option_index: 9