options: Statistics & Probability: students learn to analyze and produce data distributions, and build understanding of univariate and bivariate data. They learn to interpret data, make statistical inferences, justify conclusions, understand rules of probability, and use probability to make decisions.

Number & Operations in Base Ten: students learn to work with the base-ten system and build place value understanding.

Measurement & Data: students learn to work with data and measure attributes such as time, money, length, area, and volume. They may learn to compare measurements to operations and convert between different units of measure.

Operations & Algebra: students learn to solve problems using algebraic thinking and operations such as addition, subtraction, multiplication, and division. They may learn to identify and explain arithmetic patterns, evaluate or manipulate numerical expressions, and reason with equations and inequalities.

Modeling: students learn to choose and use appropriate mathematics and statistics to analyze real-world empirical situations, improve decisions, and report on their conclusions and the reasoning behind them.

Number & Operations - Fractions: students learn to understand fractions as numbers and may work with them using addition, subtraction, multiplication, and division.

Geometry: students learn to classify geometric figures by their properties, and understand and compare the relationships between them. Students develop and use formulas to compute lengths, areas, and volumes, and they use transformations to generate new shapes and compare existing ones. Geometry can be studied with and without coordinates.

Counting & Cardinality: students learn to know number names and the count sequence, count to tell the number of objects, and compare numbers.

Ratios & Proportional Relationships: students learn to recognize, describe, represent, and reason with ratios, rates, proportional relationships, and percent.

Number Systems and Quantity: students learn to understand the complex number system and reason quantitatively.

Functions: students learn to define, use, and evaluate functions to model relationships between quantities.
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.
[IMG]
\(P(\text{the card is a queen})\)
\(P(\text{the card is a heart})\)
\(P(\text{the card is a queen and heart})\)
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?” (
\(P(\text{the card is an 8| the card is a spade})\)
).


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