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:
{
  "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:
{
  "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:
{
  "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: Reason quantitatively and use units to solve problems.

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

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

Use complex numbers in polynomial identities and equations.

Compute fluently with multi-digit numbers and find common factors and multiples.

Perform operations on vectors.

Use properties of rational and irrational numbers.

Perform operations on matrices and use matrices in applications.

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

Represent and model with vector quantities.

Represent complex numbers and their operations on the complex plane.

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

Extend the properties of exponents to rational exponents.

Perform arithmetic operations with complex numbers.
problem_activity: Activity
The purpose of this activity is for students to practice interpreting the solution of an inequality, especially those involving negative numbers. Students also practice representing inequalities written in natural language with symbols and number lines, and connecting the solution of an inequality with a number line. The work in this activity will be useful when students solve more complex inequalities in the associated Algebra 1 lesson.
Student Facing
Solve each inequality. Then, check your answer using a value that makes your solution true.
\(\text-2x<4\)
Solve the inequality.
Check your answer using a value that makes your solution true.
\(3x+5>6x-4\)
Solve the inequality.
Check your answer using a value that makes your solution true.
\(\text-3(x+1)\geq13\)
Solve the inequality.
Check your answer using a value that makes your solution true.
For each statement:
Use a number line to show which values satisfy the inequality.
Express the statement symbolically with an inequality.
The elevator can lift up to 1,200 pounds. Let
\(x\)
represent the weight being lifted by the elevator.
[IMG]
Over the course of the senator's term, her approval rating was always around 53% ranging 3% above or below that value. Let
\(x\)
represent the senator’s approval rating.
[IMG]
There's a minimum of 3 years of experience required. Let
\(x\)
represent the years of experience a candidate has.
[IMG]
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Activity Synthesis
The goal of this discussion is to help students make sense of their solutions. Here are questions for discussion:
"After you checked your answers using a value from your solution, did any of your answers change?" (Yes. Initially I thought the solution was
\(x <\text-2\)
, but I used
\(\text-3\)
to check my answer and realized it was incorrect. When I substituted
\(\text-3\)
for
\(x\)
in the original inequality, I got
\(6<4\)
which is not true. This reminds me to change the inequality symbol when I multiply or divide by a negative number.)
"Could the elevator hold 1,300 pounds?" (No. It holds
up to
1,200, so this is the largest amount the elevator can hold.)
"Could the senator’s approval rating be 54%?" (Yes. The situation says the approval rating can be 3 points above and below 53%, and 54% fits that requirement.)
"How can the number line be useful in writing the solution to an inequality?" (It helps to see which values are included in the solution and which are not. It can also be used to check your answer after you solve an inequality.)
"On the graph of the minimum years of experience for a candidate, does it make sense to have an upper bound? What examples can you think of that would make this situation have an upper bound?" (Yes, because realistically a person cannot have infinity years of experience and that is what the current graph of the inequality reprsents. For example, many people retire after a certain age, so it would be hard for someone to have more than 50 years of experience. Also, the average lifespan is about 110 and people ususally start working as teens or young adults, so this creates another limit on how many years of experience someone can actually have.)


correct_option_index: 1