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: Use functions to model relationships between quantities.

Construct and compare linear and exponential models and solve problems.

Understand the concept of a function and use function notation.

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

Build new functions from existing functions.

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

Define, evaluate, and compare functions.

Model periodic phenomena with trigonometric functions.

Extend the domain of trigonometric functions using the unit circle.

Analyze functions using different representations.

Build a function that models a relationship between two quantities.

Prove and apply trigonometric identities.
problem_activity: Activity
The purpose of this activity is for students to work with a function where either variable could be the independent variable. Knowing the total value for an unknown number of dimes and quarters, students are first asked to consider if the number of dimes could be a function of the number of quarters and then asked if the reverse is also true. Since this isn't always the case when students are working with functions, the discussion should touch on reasons for choosing one variable vs. the other, which can depend on the types of questions one wants to answer.
Identify students who efficiently rewrite the original equation in the third problem and the last problem to share during the discussion.
Launch
Arrange students in groups of 2. Give students 3–5 minutes of quiet work time followed by partner discussion for students to compare their answers and resolve any differences. Follow with a whole-class discussion.
Representation: Internalize Comprehension.
Provide appropriate reading accommodations and supports to ensure students access to written directions, word problems and other text-based content.
Supports accessibility for: Language; Conceptual processing
Reading: MLR6 Three Reads.
Use this routine to support reading comprehension and to set students up to interpret the representations of the situation provided in the task statement (an equation) and discussion (function diagrams). In the first read, students read the information with the goal of comprehending the situation (e.g., Jada has dimes and quarters). In the second read, ask students to identify important quantities. Listen for, and amplify, naming of the quantities that vary in relation to each other in this situation: number of dimes, total value of dimes, number of quarters, total value of quarters. After the third read, ask students to discuss possible strategies to answer the questions that follow, paying attention to the different coin values, and how each question is phrased. This will help students comprehend the problem and make sense of important quantities and variables when working with a function in which either variable could be the independent variable.
Design Principle(s): Support sense-making
Student Facing
Jada had some dimes and quarters that had a total value of
$
12.50. The relationship between the number of dimes,
\(d\)
, and the number of quarters,
\(q\)
, can be expressed by the equation
\(0.1d + 0.25q = 12.5\)
.
If Jada has 4 quarters, how many dimes does she have?
If Jada has 10 quarters, how many dimes does she have?
Is the number of dimes a function of the number of quarters? If yes, write a rule (that starts with
\(d = \)
...) that you can use to determine the output,
\(d\)
, from a given input,
\(q\)
. If no, explain why not.
If Jada has 25 dimes, how many quarters does she have?
If Jada has 30 dimes, how many quarters does she have?
Is the number of quarters a function of the number of dimes? If yes, write a rule (that starts with
\(q=\)
...) that you can use to determine the output,
\(q\)
, from a given input,
\(d\)
. If no, explain why not.
Student Response
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Anticipated Misconceptions
Some students may be unsure how to write rules for the number of dimes as a function of the number of quarters and vice versa. Prompt them to use the provided equation and what they know about keeping equations equal to create the new equations.
Activity Synthesis
Select previously identified students to share their rules for dimes as a function of the number of quarters and quarters as a function of the number of dimes, including the steps they used to rewrite the original equation.
Tell students that if we write an equation like
\(d = 125 - 2.5q\)
, this shows that
\(d\)
is a function of
\(q\)
because it is clear what the output (value for
\(d\)
) should be for a given input (value for
\(q\)
).
Display the diagrams for all to see:
[IMG]
When we have an equation like
\(0.1d + 0.25q = 12.5\)
, we can choose either
\(d\)
or
\(q\)
to be the independent variable. That means we are viewing one as depending on the other. If we know the number of quarters and want to answer a question about the number of dimes, it is helpful to write
\(d\)
as a function of
\(q\)
. If we know the number of dimes and want to answer a question about the number of quarters, it is helpful to write
\(q\)
as a function of
\(d\)
.
Ensure students understand that we can’t always do this type of rearranging with equations and have it make sense because sometimes only one variable is a function of the other, and sometimes neither is a function of the other. For example, students saw earlier that while squaring values is a function, the reverse—that is, identifying what value was squared—is not. We will continue to explore when these different things happen in future lessons.


correct_option_index: 6