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: Work with numbers 11–19 to gain foundations for place value.

Generalize place value understanding for multi-digit whole numbers.
Grade 4 expectations in this domain are limited to whole numbers less than or equal to 1,000,000.

Extend the counting sequence.

Use place value understanding and properties of operations to perform multi-digit arithmetic. (A range of algorithms may be used.)

Use place value understanding and properties of operations to perform multi-digit arithmetic.
Grade 4 expectations in this domain are limited to whole numbers less than or equal to 1,000,000.

Perform operations with multi-digit whole numbers and with decimals to hundredths.

Understand the place value system.

Use place value understanding and properties of operations to add and subtract.

Understand place value.
problem_activity: Narrative
The purpose of this activity is for students to practice using the subtraction algorithm introduced in a previous lesson. Provide base-ten blocks for students who choose to use them to support their reasoning about the algorithm.
MLR8 Discussion Supports:
Synthesis: Before students share their reasoning, remind them to use words such as decompose, ones, tens, and hundreds.
Advances: Speaking, Representing
Engagement: Provide Access by Recruiting Interest.
Leverage choice around perceived challenge. Invite students to select at least 3 of 5 problems to complete.
Supports accessibility for: Organization, Attention, Social-emotional skills
Required Materials
Materials to Gather
Base-ten blocks
Launch
Groups of 2
Display Kiran’s algorithm from the previous lesson.
“Here’s a subtraction algorithm you saw in an earlier lesson. What might be the first thing you’d do if you are to use this algorithm to find the value of the subtraction expressions in the activity?” (Write the numbers in expanded form and stack them.)
1 minute: quiet think time
Share responses.
Give students access to base-ten blocks.
Activity
“Take some quiet time to try this algorithm. Check in with your partner if you have questions.”
5–7 minutes: independent work time
If students have questions about the notation used to record the decomposition of a hundred or ten into more tens or ones, consider asking:
“Is there any place in the problem where you don’t have enough tens or ones?”
“How could you get more tens (or ones)?”
“How could you record a hundred being decomposed into 10 tens (or a ten decomposed into 10 ones)?”
Student Facing
Here is a subtraction algorithm you saw in an earlier lesson:
[IMG]
Try using this algorithm to find the value of each difference. Show your reasoning. Organize it so it can be followed by others.
\(283 - 159\)
\(425 - 192\)
\(639 - 465\)
\(591 - 128\)
\(832 - 575\)
Student Response
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Advancing Student Thinking
If students do not record the multiple decompositions in the last problem, consider asking:
“What units did you need to decompose in this problem?”
“Where would it make the most sense to you to record how you decomposed a hundred into more tens? A ten into more ones?”
Activity Synthesis
Select students to share their reasoning for 2–3 problems. Choose problems to focus on based on common questions that came up. Be sure to discuss the last problem, which requires decompositions of both a ten and a hundred.
For the last problem, ask: “How did you decide how to record the ten and hundred that needed to be decomposed?” (I started subtracting the ones and decomposed a ten into 10 ones so I had already crossed off the 30 and written 20. When I decomposed the hundred into tens, I just decided to write the 120 on top of the 20 like I wrote the 20 on top of the 30.)
Record the completed algorithm, showing the decompositions of tens and hundreds.
Consider asking:
“Where were there not enough tens or ones to subtract?”
“What was decomposed and how was it recorded?”
“Did you notice any places where you might have made the error we saw in Lin's work?"


correct_option_index: 3