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: Prove and apply trigonometric identities.

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

Use functions to model relationships between quantities.

Analyze functions using different representations.

Construct and compare linear and exponential models and solve problems.

Build new functions from existing functions.

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

Model periodic phenomena with trigonometric functions.

Build a function that models a relationship between two quantities.

Define, evaluate, and compare functions.

Extend the domain of trigonometric functions using the unit circle.

Understand the concept of a function and use function notation.
problem_activity: Optional activity
This optional activity gives students an additional opportunity to apply what they know about key characteristics of an exponential function and use it to model real-world data. Unlike previous activities, the data have not been adjusted to perfectly fit an exponential model. Students will need to make estimates and think strategically about what model to choose. Because the years chosen for data points are close together, students may use repeated multiplication to answer the questions rather than writing and evaluating an expression that models the exponential function.
The activity also allows students to practice articulating their reasoning process. The info gap structure requires students to determine what information is necessary to solve a problem and then to ask for the information. This may take several rounds of discussion, as their first requests may not yield the necessary information (MP1). It also allows students to refine the language they use and ask increasingly more precise questions until they get the information they need (MP6).
Here is the text of the cards for reference and planning:
[IMG]
In Problem Card 1, answers will vary based on the data points and the strategy students use to find the growth factor. Strategies to watch for include:
Estimate that the phone sales are roughly doubling each year and use that to estimate future sales
Calculate a growth factor using the data from two consecutive years
Find the average of the growth factors between 2008 and 2009 and between 2009 and 2010
For the second problem, the third approach will not arise because only two sales numbers are available on the data card.
Launch
Tell students they will continue to work with interpreting exponential relationships. Ask students “How many of you have a smartphone?” Give students a brief overview on the first appearance of smartphones and when they became widely popular (around 2007). Explain that annual records of smartphone sales are available starting from 2008 until now.
Explain the info gap structure, and consider demonstrating the protocol if students are unfamiliar with it.
Arrange students in groups of 2. In each group, distribute a problem card to one student and a data card to the other student. After reviewing students' work on the first problem, give them the cards for a second problem and instruct partners to switch roles.
Conversing:
This activity uses
MLR4 Information Gap
to give students a purpose for discussing information necessary to solve problems involving equivalent ratios. Consider providing questions or question starters for students who need a starting point such as: “Can you tell me . . .”, and “Why do you need to know . . .?”
Design Principle(s): Cultivate Conversation
Engagement: Develop Effort and Persistence.
Display or provide students with a physical copy of the written directions. Check for understanding by inviting students to rephrase directions in their own words. Keep the display of directions visible throughout the activity.
Supports accessibility for: Memory; Organization
Student Facing
Your teacher will give you either a problem card or a data card. Do not show or read your card to your partner.
If your teacher gives you the data card:
Silently read the information on your card.
Ask your partner “What specific information do you need?” and wait for your partner to ask for information. Only give information that is on your card. (Do not figure out anything for your partner!)
Before telling your partner the information, ask “Why do you need to know (that piece of information)?”
Read the problem card, and solve the problem independently.
Share the data card, and discuss your reasoning.
If your teacher gives you the problem card:
Silently read your card and think about what information you need to answer the question.
Ask your partner for the specific information that you need.
Explain to your partner how you are using the information to solve the problem.
When you have enough information, share the problem card with your partner, and solve the problem independently.
Read the data card, and discuss your reasoning.
Pause here so your teacher can review your work. Ask your teacher for a new set of cards and repeat the activity, trading roles with your partner.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Activity Synthesis
Invite students to share different ways they estimated a growth factor for smartphone sales, including:
using a rough estimate of 2
using sales data from two consecutive years (2008 and 2009, or 2009 and 2010)
using sales data from all three years and taking an average of growth factors for consecutive years
The first of these methods is efficient and allows for quick calculations. It is not as accurate, however, as the other estimates and may lead to a less reliable prediction. In this case, it leads to an overestimate of phone sales since the actual growth factors are less than 2.
Consider showing a model and a graph generated using a calculator or graphing technology. Based on this model, about 125 million of that brand of smartphones were sold in 2012.
[IMG]
Next, focus the whole-class discussion on the limitations of a real-world model. In this context, market saturation limits the growth of the cell phone sales. Discuss questions such as:
“According to your model, how many smartphones were sold worldwide in 2018?” (About 5 billion.)
“The world population in 2018 is about 7.6 billion. Is the number of smartphones sold based on your model realistic? Why or why not?” (No.  Allowing for children, the elderly, and underdeveloped countries, 66% of the population buying one brand of smartphone is not realistic.)
“Could the sales of smartphones continue to grow exponentially? Why or why not?” (No. There is a finite number of people who will buy a particular brand of smartphone. The data should level off, with surges when new models come out.)


correct_option_index: 4