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 and apply properties of operations and the relationship between addition and subtraction.

Represent and solve problems involving multiplication and division.

Represent and solve problems involving addition and subtraction.

Write and interpret numerical expressions.

Create equations that describe numbers or relationships

Solve problems involving the four operations, and identify and explain patterns in arithmetic.

Apply and extend previous understandings of arithmetic to algebraic expressions.

Analyze and solve linear equations and pairs of simultaneous linear equations.

Multiply and divide within 100.

Add and subtract within 20.

Understand the connections between proportional relationships, lines, and linear equations.

Understand solving equations as a process of reasoning and explain the reasoning.

Represent and analyze quantitative relationships between dependent and independent variables.

Write expressions in equivalent forms to solve problems.

Solve equations and inequalities in one variable.

Analyze patterns and relationships.

Work with equal groups of objects to gain foundations for multiplication.

Work with radicals and integer exponents.

Understand the relationship between zeros and factors of polynomials.

Work with addition and subtraction equations.

Use properties of operations to generate equivalent expressions.

Perform arithmetic operations on polynomials.

Represent and solve equations and inequalities graphically.

Generate and analyze patterns.

Solve systems of equations.

Use the four operations with whole numbers to solve problems.

Use polynomial identities to solve problems.

Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.

Gain familiarity with factors and multiples.

Understand properties of multiplication and the relationship between multiplication and division.

Interpret the structure of expressions.

Reason about and solve one-variable equations and inequalities.

Solve real-life and mathematical problems using numerical and algebraic expressions and equations.

Rewrite rational expressions.
problem_activity: Narrative
In this activity, students work with customary units of capacity for liquids (gallon, quart, and cup). The task prompts them to discern the relationship between the units, express the relationships with multiplicative comparison statements, and perform conversions to solve problems. When they identify the relationship between gallons, quarts, and cups students reason abstractly and quantitatively (MP2).
To compare quantities given in different units, students choose which unit to use for comparison and consider the implications of their choice. Students are likely to convert a value in a larger unit to to a smaller unit (from gallons and quarts to cups), but some may reason the other way around (from cups and quarts to gallons, ending up with fractional amounts). In grade 4, students are expected only to convert from a larger unit to a smaller one.
MLR8 Discussion Supports.
To support the transfer of new vocabulary to long term memory, invite students to chorally repeat these words or phrases in unison 1-2 times: gallon, quart, cup.
Advances: Representing, Speaking, Listening
Representation: Access for Perception.
Give students access to connecting cubes. Invite them to use the cubes to support their thinking in a way that makes sense to them. If students need additional support, invite them to use the cubes to first build and label a representation of one cup, one quart, and one gallon.
Supports accessibility for: Conceptual Processing
Required Materials
Materials to Gather
Containers of different sizes
Required Preparation
Gather a one-gallon jug (with or without milk), a one-quart container, and a one-cup container for display during the launch.
On chart paper, create the table in the activity with an extra column for showing the amounts of lassi in cups, to be displayed during synthesis.
Launch
Groups of 2
Read the first sentence and the bullet points in the first problem.
Display a one-gallon jug (with or without milk), a one-quart container, and a one-cup container. Tell students that, when filled up, the jug holds 1 gallon.
“Here is a container that holds 1 quart of liquid. How is that amount related to the amount in the jug?” (It takes 4 of these to fill up the jug.)
“Here is a container that holds 16 cups. How is that amount related to the amount in the jug?” (It takes 16 of these to fill up the jug.)
“Complete the statements in the first problem. Then, share your responses with your partner.“
1 minute: quiet think time
1 minute: partner discussion
“Let’s use what we know about these units for measuring liquids to solve some problems.”
Explain to students that mango lassi is an Indian drink, a smoothie made of mango and yogurt or milk.
Activity
5 minutes: independent work time
5 minutes: partner discussion
Monitor for:
which unit(s) students use to compare the amounts
the representations students use to reason about the quantities
Student Facing
Here are three true statements about this jug of milk.
This jug contains 1 gallon of milk.
This jug contains 4 quarts of milk.
This jug contains 16 cups of milk.
Complete each statement so that it is true:
[IMG]
One gallon is __________ times as much as 1 quart.
One gallon is __________ times as much as 1 cup.
One quart is __________ times as much as 1 cup.
For a potluck party, Priya and three other relatives are bringing mango lassi.
[IMG]
|  guest  | amount of lassi |
|---------|-----------------|
|  Priya  |     10 cups     |
|  Uncle  |    3 quarts     |
| Cousin  |     8 cups      |
| Grandma |    2 gallons    |
Who prepared the most mango lassi? Explain or show your reasoning.
How many cups of lassi did all the guests bring?
Complete this sentence: Priya’s grandma made __________ times as much lassi as Priya’s cousin. Show how you know.
Student Response
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Advancing Student Thinking
Students may compare only the numbers in the table and not the units. Encourage them to compare two amounts at a time—for instance, 8 cups and 2 gallons. Ask them how they would know which of the two is greater. To draw their attention to the units, consider asking them to illustrate or describe those two amounts using actual containers (if available) or using drawings.
Activity Synthesis
Poll the class on the unit they used to compare the amounts of lassi.
Select a student who chose each option to share the rationale behind their decision.
Display the table with an extra column for showing the amounts in cups.
Invite students to share their responses to the questions about lassi and their reasoning, completing the table along the way.
|  guest  | amount of lassi | amount of lassi (cups) |
|---------|-----------------|------------------------|
|  Priya  |     10 cups     |           10           |
|  Uncle  |    3 quarts     |           12           |
| Cousin  |     8 cups      |           8            |
| Grandma |    2 gallons    |           32           |
Highlight that converting the amounts into the same unit helped us: see the greatest amount, find the total amount, and see how many times as much lassi one guest made compared to another guest.
To reinforce students' understanding of the relationship between the units of capacity used here, consider creating a diagram such as shown.
[IMG]
“How does the diagram show that 1 gallon is 4 times as much as 1 quart and 16 times as much as 1 cup?”


correct_option_index: 25