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.


Standard options: 
A. Represent addition and subtraction with objects, fingers, mental images, drawings (Drawings need not show details, but should show the mathematics in the problem. (This applies wherever drawings are mentioned in the Standards.)), sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations.
B. For any number from 1 to 9, find the number that makes 10 when added to the given number, e.g., by using objects or drawings, and record the answer with a drawing or equation.
C. Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or drawings, and record each decomposition by a drawing or equation (e.g., 5 = 2 + 3 and 5 = 4 + 1).
D. Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.
E. Fluently add and subtract within 5.


Thought: This task involves representing addition with objects such as number cards and dice.
Answer: A

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
$$-\frac{1}{4}$$


Standard options: 
A. Solve real-world and mathematical problems involving the four operations with rational numbers. (Computations with rational numbers extend the rules for manipulating fractions to complex fractions.)
B. Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.
C. Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.


Thought: This problem involves understanding fractions and division to divide rational numbers, such as 1, -4, 4, and -1. This problem also involves an understanding of multiplication: if p and q are integers, then -(p/q) = (-p)/q = p/(-q).
Answer: B

Example 3:
Problem/activity:
Find the general solution for:
$${\mathrm{cos}\left({\pi\over2}+x\right)=\mathrm{tan}{\pi\over4}}$$


Standard options: 
A. Prove the Pythagorean identity sin^(2)(θ) + cos^(2)(θ) = 1 and use it to find sin(θ), cos(θ),or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.
B. Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.


Thought: An initial step when solving for x in this problem requires the use of the addition formula for cosine on the left side of the equation. Solving this problem does not require use of the Pythagorean identity.
Answer: B

Now, assign the following problem or activity to one or more relevant standards it aligns with.
shot_info: im_task_001477

K

fl_problem_001097

7

fl_problem_001894

HS_0
options: Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B.

Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results.

Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer.

Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”).

Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.
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
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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: 0

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