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: Apply and extend previous understandings of multiplication and division to multiply and divide fractions.

Extend understanding of fraction equivalence and ordering.
Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.

Develop Understanding of fractions as numbers.
Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8.

Understand decimal notation for fractions, and compare decimal fractions.
Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.

Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.
Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.

Use equivalent fractions as a strategy to add and subtract fractions.
problem_activity: Narrative
The purpose of this activity is to represent and solve a measurement problem with fractions. Students may approach this activity in multiple ways and are invited to apply what they know about operations with fractions, comparing fractions, and fraction equivalence to make sense of and solve the problems (MP2). Throughout the activity, listen for the ways students use what they know about comparing fractions and fraction equivalence as they make sense of the problem. Although students may consider ways to subtract fractions with unlike denominators, this is not a requirement for grade 4. Focus the conversation during the activity and the synthesis on how students can solve the problem by reasoning about equivalent fractions and the representation they use to make sense of the problem.
MLR8 Discussion Supports.
Synthesis: Create a visual display of the diagrams. As students share their strategies, annotate the display to illustrate connections. For example, next to each representation, write how the diagram relates to the situation.
Advances: Speaking, Representing
Launch
Groups of 2
Activity
5 minutes: independent work time
5 minutes: partner work
Monitor for different representations of the situation (for example, equations, drawings, and number lines).
Student Facing
In science class, Noah, Tyler, and Jada are building a model of a roller coaster out of 1-foot-long paper straws.
Noah needs a piece that is
\(\frac{7}{12}\)
foot long.
Tyler needs one that is
\(\frac{1}{4}\)
foot long.
Jada needs one that is shorter than the other two.
Jada says, “We can just use one straw for all these pieces.”
[IMG]
Draw a diagram to represent this situation and explain to your partner how it matches the situation. Then, find the length of the piece of straw that could be Jada’s piece.
Did Noah use more than
\(\frac{1}{2}\)
foot or less than
\(\frac{1}{2}\)
foot of straw? Explain or show your reasoning.
Tyler says, “If Jada uses a piece that is
\(\frac{1}{6}\)
foot long, there would be a piece of straw that is
\(\frac{1}{12}\)
foot left.”
Do you agree or disagree with Tyler? Explain your reasoning.
Student Response
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Advancing Student Thinking
If students create representations that do not match the quantities in the problem, consider asking:
“How does your representation show the length of the straw? How does it show Noah and Tyler’s pieces?”
“How does your representation show the action in the situation?”
Activity Synthesis
Invite previously selected students to share their arguments and allow peers to extend and add ideas to the conversation.
“How does each representation match this situation about the straw?” (They each show the total length of the straw and ways it could be broken into smaller parts. They each show the length of Noah and Tyler’s pieces. They show how long Jada’s piece could be.)


correct_option_index: 1

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