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: Draw informal comparative inferences about two populations.

Develop understanding of statistical variability.

Make inferences and justify conclusions from sample surveys, experiments, and observational studies.

Use probability to evaluate outcomes of decisions.

Summarize and describe distributions.

Interpret linear models.

Calculate expected values and use them to solve problems.

Investigate patterns of association in bivariate data.

Summarize, represent, and interpret data on a single count or measurement variable.

Understand and evaluate random processes underlying statistical experiments.

Investigate chance processes and develop, use, and evaluate probability models.

Understand independence and conditional probability and use them to interpret data.

Use the rules of probability to compute probabilities of compound events in a uniform probability model.

Use random sampling to draw inferences about a population.

Summarize, represent, and interpret data on two categorical and quantitative variables.
problem_activity: Activity
The mathematical purpose of this activity is to use a two-way table as a sample space to decide if events are independent and to estimate conditional probabilities. Listen for students mentioning the concept of conditional probability.
Launch
Arrange students in groups of two. Give students quiet time to work the questions, have partners compare answers, and then have a whole-class discussion.
Speaking, Representing: MLR8 Discussion Supports.
Use this routine to support small group discussion. After students have completed a large portion of the task prompts, provide students with sentence frames to support their conversation comparing their answers and determining if they are correct. For example, "I agree because . . .”, "I disagree because . . .”, “Why do you think . . . ?” This will help students understand how to use probabilities to describe dependent and independent events.
Design Principle(s): Support sense-making
Student Facing
A suspected cause of a disease is a variation in a certain gene. A study gathers at-risk people at random and tests them for the disease as well as for the genetic variation.
|                                     | has the disease | does not have the disease |
|-------------------------------------|-----------------|---------------------------|
|      has the genetic variation      |                 |                           |
| does not have the genetic variation |                 |                           |
A person from the study is selected at random. Let A represent the event “has the disease” and B represent “has the genetic variation.”
Use the table to find the probabilities. Show your reasoning.
\(P(\text{A})\)
\(P(\text{B})\)
\(P(\text{A and B})\)
\(P(\text{A | B})\)
Based on these probabilities, are the events independent? Explain your reasoning.
A company that tests for this genetic variation has determined that someone has the variation and wants to inform the person that they may be at risk of developing this disease when they get older. Based on this study, what percentage chance of getting the disease should the company report as an estimate to the person? Explain your reasoning.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Student Facing
Are you ready for more?
Find or collect your own data that can be represented using a two-way table. Some ideas include finding data about a sports team (shooting percentage greater than 50% or less than 50% vs. winning or losing) or clinical trials for a medication (placebo or medication vs. symptom or no symptom).
Create a two-way table to represent the sample space.
Use probabilities to determine whether events summarized in the table are dependent or independent events.
When data is collected in an experiment or a survey it is often difficult to show evidence that events are dependent. Why do you think this occurs? Explain your reasoning.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Activity Synthesis
The purpose of this discussion is for students to describe how to use probabilities to describe dependent and independent events.
Here are some questions for discussion.
“What is one way to determine whether 'having the disease' and 'having the genetic variation' are dependent or independent events?” (You find the conditional probability of having the disease given the condition of having the genetic variations, and then compare it to the probability of having the disease. If they are equal, they are independent.)
“What is another way?” (You find the probability of having the disease and having the genetic variations, and compare that to the product of the probability of having the disease and the probability of having the genetic variation.)
“What other information would it be helpful for the company to report?” (It would be helpful to know the overall risk of getting the disease in addition to knowing the risk of getting it if you have the genetic variation.)


correct_option_index: 11