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.


Your response, in json format:
{
  "explanation": "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.


Your response, in json format:
{
  "explanation": "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.


Your response, in json format:
{
  "explanation": "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: Represent complex numbers and their operations on the complex plane.

Extend the properties of exponents to rational exponents.

Apply and extend previous understandings of numbers to the system of rational numbers.

Know that there are numbers that are not rational, and approximate them by rational numbers.

Use complex numbers in polynomial identities and equations.

Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.

Perform operations on vectors.

Perform arithmetic operations with complex numbers.

Perform operations on matrices and use matrices in applications.

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

Reason quantitatively and use units to solve problems.

Use properties of rational and irrational numbers.

Represent and model with vector quantities.

Compute fluently with multi-digit numbers and find common factors and multiples.
problem_activity: Activity
The purpose of this activity is to understand that the division facts for rational numbers are simply a consequence of the multiplication done previously. Students work several numerical examples relating multiplication to division, and then articulate a rule for the sign of a quotient based on the signs of the dividend and divisor (MP8).
Monitor for students who identify and describe the rule clearly.
Launch
Remind students that we can rearrange division equations to be multiplication equations, and vice versa. It may be useful to demonstrate with positive numbers if students struggle to recall this. For example, ask how we could rewrite
\(10 \div 2 = 5\)
as a multiplication equation. Students can also reference the multiplication table from a previous lesson, if needed.
Arrange students in groups of 2. Give students 4 minute of quiet work time followed by 2 minutes of partner discussion, then follow with whole-class discussion.
Engagement: Develop Effort and Persistence.
Connect a new concept to one with which students have experienced success. For example, reference examples from the previous lesson on signed multiplication to provide an entry point into this activity.
Supports accessibility for: Social-emotional skills; Conceptual processing
Reading: MLR6 Three Reads.
Use this routine to support students’ reading comprehension of the final question. In the first read, students read the text with the goal of understanding what the situation is about (e.g., Han and Clare are walking toward each other). In the second read, ask students to name the important quantities (e.g., Han’s velocity, Clare’s velocity, each person's distance from 0 in feet, elapsed time in seconds). After the third read, ask students to brainstorm possible strategies to determine where each person will be 10 seconds before they meet up, and when each person will be
\(-10\)
feet from the meeting place. This will help students connect the language in the word problem with the reasoning needed to solve the problem.
Design Principle(s): Support sense-making
Student Facing
Find the missing values in the equations
\(\text-3 \boldcdot 4 = \text{?}\)
\(\text-3 \boldcdot \text{?} = 12\)
\(3 \boldcdot \text{?} = 12\)
\(\text{?} \boldcdot \text-4 = 12\)
\(\text{?} \boldcdot 4 = \text-12\)
Rewrite the unknown factor problems as division problems.
Complete the sentences. Be prepared to explain your reasoning.
The sign of a positive number divided by a positive number is always:
The sign of a positive number divided by a negative number is always:
The sign of a negative number divided by a positive number is always:
The sign of a negative number divided by a negative number is always:
Han and Clare walk towards each other at a constant rate, meet up, and then continue past each other in opposite directions. We will call the position where they meet up 0 feet and the time when they meet up 0 seconds.
Han's velocity is 4 feet per second.
Clare's velocity is -5 feet per second.
Where is each person 10 seconds before they meet up?
When is each person at the position -10 feet from the meeting place?
Student Response
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Student Facing
Are you ready for more?
It is possible to make a new number system using
only
the numbers 0, 1, 2, and 3. We will write the symbols for multiplying in this system like this:
\(1 \otimes 2 = 2\)
. The table shows some of the products.
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "row"}, "span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "43618"}, "_value": "\\(\\otimes\\)"}]}, {"_attributes": {"scope": "col"}, "_value": "0"}, {"_attributes": {"scope": "col"}, "_value": "1"}, {"_attributes": {"scope": "col"}, "_value": "2"}, {"_attributes": {"scope": "col"}, "_value": "3"}]}]}], "tbody": [{"tr": [{"th": [{"_attributes": {"scope": "row"}, "_value": "0"}], "td": [{"_value": "0"}, {"_value": "0"}, {"_value": "0"}, {"_value": "0"}]}, {"th": [{"_attributes": {"scope": "row"}, "_value": "1"}], "td": [{}, {"_value": "1"}, {"_value": "2"}, {"_value": "3"}]}, {"th": [{"_attributes": {"scope": "row"}, "_value": "2"}], "td": [{}, {}, {"_value": "0"}, {"_value": "2"}]}, {"th": [{"_attributes": {"scope": "row"}, "_value": "3"}], "td": [{}, {}, {}, {}]}]}]}]}
In this system,
\(1 \otimes 3 = 3\)
and
\(2 \otimes 3 = 2\)
. How can you see that in the table?
What do you think
\(2 \otimes 1\)
is?
What about
\(3\otimes 3\)
?
What do you think the solution to
\(3\otimes n = 2\)
is?
What about
\(2\otimes n = 3\)
?
Student Response
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for free access to Extension Student Response.
Anticipated Misconceptions
Have students refer to their previous work with signed multiplication to help them with the division statements.
Activity Synthesis
Select students who can articulate and explain what they have learned about the sign of a quotient based on the signs of the dividend and divisor (note that students do not need to use these words, they can just give examples). Highlight that the division facts are simply a consequence of the multiplication done previously. Moving forward, we are going to take these facts to be true for all rational numbers.


correct_option_index: 5