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 the four operations with whole numbers to solve problems.

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

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

Perform arithmetic operations on polynomials.

Generate and analyze patterns.

Work with radicals and integer exponents.

Represent and solve equations and inequalities graphically.

Create equations that describe numbers or relationships

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

Add and subtract within 20.

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

Work with addition and subtraction equations.

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

Reason about and solve one-variable equations and inequalities.

Solve systems of equations.

Gain familiarity with factors and multiples.

Understand the relationship between zeros and factors of polynomials.

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

Represent and analyze quantitative relationships between dependent and independent variables.

Represent and solve problems involving addition and subtraction.

Rewrite rational expressions.

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

Interpret the structure of expressions.

Write expressions in equivalent forms to solve problems.

Analyze patterns and relationships.

Multiply and divide within 100.

Use polynomial identities to solve problems.

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

Solve equations and inequalities in one variable.

Write and interpret numerical expressions.

Use properties of operations to generate equivalent expressions.

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

Represent and solve problems involving multiplication and division.

Apply and extend previous understandings of arithmetic to algebraic expressions.
problem_activity: Narrative
The purpose of this activity is for students to revisit stage 3 of the Shake and Spill center, introduced in kindergarten. In this stage, students see a quantity broken into two parts in different ways. Student write equations to represent each decomposition. Students may write an equation in any way they choose, but the number of counters is presented first to encourage students to write the total before the equal sign. This activity builds toward a future lesson in which students solve Put Together/Take Apart, Addend Unknown story problems and write equations to match them.
During this activity, the teacher collects and displays different equations that students write for the first round. This includes equations where the total is before the equal sign, such as
\(9 = 7 + 2\)
. During the synthesis, students are encouraged to think about how an equation with the total before the equal sign relates back to the context of playing the game (MP2).
Required Materials
Materials to Gather
Cups
Two-color counters
Materials to Copy
Shake and Spill Stage 3 Recording Sheet Grade 1
Required Preparation
Each group of 2 needs 10 two-color counters and a cup (at least 8 oz).
Launch
Groups of 2
Give each group a cup, 10 two-color counters, and recording sheets.
"Today we will revisit a game you played in kindergarten called Shake and Spill. Let's play one round together to make sure everyone remembers how to play."
Display two-color counters and the cup.
“I have some two-color counters. Let’s count them together.”
Place counters in the cup as you count aloud.
“I’m going to write 6 under Total Number of Counters."
Demonstrate shaking and spilling the counters.
"How many red counters are there? How many yellow counters are there?”
30 seconds: quiet think time
Record responses in the table.
“What equation can we write to match the counters?” (
\(4 + 2 = 6\)
,
\(6 = 4 + 2\)
,
\(2 + 4 = 6, 6 = 2 + 4\)
)
30 seconds: quiet think time
30 seconds: partner discussion
Share and record responses.
If needed, play another round.
Activity
“Play the game with your partner. For the first game, you will use 9 counters and record in your book. After the first game, you may choose the number of counters that you want to use, and record on the separate recording sheet.”
10 minutes: partner work time
If needed, ask “Is there another equation you can write to show this round?”
Monitor for and collect 5–6 combinations and equations from round 1.
Student Facing
| total number of counters | red                   counters | yellow                   counters | equations |
|--------------------------|--------------------------------|-----------------------------------|-----------|
|                          |                                |                                   |           |
|                          |                                |                                   |           |
[IMG]
Round 1:
| total number of counters | red                   counters | yellow                   counters | equations |
|--------------------------|--------------------------------|-----------------------------------|-----------|
|            9             |                                |                                   |           |
|            9             |                                |                                   |           |
|            9             |                                |                                   |           |
|            9             |                                |                                   |           |
|            9             |                                |                                   |           |
|            9             |                                |                                   |           |
Student Response
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Activity Synthesis
Display collected combinations and equations.
“What do you notice about the equations I collected during the first round?” (There are different numbers in the equations. They all equal nine. Sometimes the total is before the equation and sometimes it is after.)
“What does the equation
\(9 = 7 + 2\)
mean?” (The nine counter total is the same amount as seven red counters and two yellow counters or seven yellow and two red.)


correct_option_index: 9