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.


Concept/skill: 
Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.

Answer: 
No. Though this task involves addition, it does not involve any word problems.

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


Concept/skill: 
Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.

Answer: 
Yes. 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).

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


Concept/skill: 
Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

Answer: 
Yes. 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.

Now, does the problem or activity shown below enable students to learn the full intent of the following concept or skill?

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fl_problem_001097_pos_0

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standard_description: Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model.
problem_activity: Warm-up
The mathematical purpose of this activity is to explore, formally define, and introduce notation for the concept of
conditional probability.
Launch
Display the image of a deck of cards.
[IMG]
Explain that there are four suits (spades, hearts, diamonds, clubs) with 13 cards in each suit (ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, jack, queen, king) for a total of 52 cards. Hearts and diamonds are always considered red cards while clubs and spades are always considered black cards. Face cards are jack, queen, or king.
Unless otherwise noted, a standard deck of cards in these materials will consist of these 52 cards.
Representation: Internalize Comprehension.
Use virtual or concrete manipulatives to connect symbols to concrete objects or values. Provide students with a standard deck of cards to see the different suits and the cards that are red and black. Allow groups of students to manipulate the cards to illustrate the probabilities in the problems.
Supports accessibility for: Visual-spatial processing; Conceptual processing
Student Facing
Noah will select 1 card at random from a standard deck of cards. Find the probabilities. Explain or show your reasoning.
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\(P(\text{the card is a queen})\)
\(P(\text{the card is a heart})\)
\(P(\text{the card is a queen and heart})\)
Elena pulls out only the hearts from the deck and sets the rest of the cards aside. She shuffles the hearts and draws one card. What is the probability she gets a queen?
Student Response
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Activity Synthesis
The purpose of this discussion is to formally define conditional probability and to introduce notation for conditional probability. Tell students that “
conditional probability
is defined as the probability of an event happening under the condition that another event occurs.” Use Elena’s situation as an example to show the notation for conditional probability:
\(P(\text{the card is a queen | the card is a heart})\)
which can be read as “the probability that the card is a queen under the condition that the card is a heart has occurred.” In this situation, Elena made sure the card would be a heart by removing the other suits and drawing only from those cards.
Display
\(P(\text{the card is a heart | the card is a queen})\)
for all to see and ask students to explain what it means, then ask students to find the probability. (It means the probability that the card is a heart under the condition that it is a queen. It has the value of
\(\frac{1}{4}\)
since there is one heart card among the 4 queens.) This is the notation we will use throughout this unit to represent conditional probability. Ask students “How would you write what is the probability that a card is an 8 under the condition that is a spade?” (
\(P(\text{the card is an 8| the card is a spade})\)
).


aligns: yes