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.

Functions: students learn to define, use, and evaluate functions to model relationships between quantities.

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.

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

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.

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

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

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

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.
problem_activity: Optional activity
The activity provides further practice in finding probabilities of events.
In this activity, students see an experiment that has two steps where the result of the first step influences the possibilities for the second step. Often this process is referred to as doing something “without replacement.” At this stage, students should approach these experiments in a very similar way to all of the other probability questions they have encountered, but they must be very careful about the number of outcomes in the sample space (MP6).
Launch
Keep students in groups of 2. Give students 5–7 minutes of quiet work time followed by partner and whole-class discussion.
Identify students who are not noticing that it is impossible to draw the same color twice based on the instructions. Refocus these students by asking them to imagine drawing a red card on the first pick and thinking about what’s possible to get for the second card.
Representation: Access for Perception.
Provide access to concrete manipulatives. Provide five different colored cards for students to view or manipulate. These hands-on models will help students identify characteristics or features, and support finding outcomes for calculating probabilities.
Supports accessibility for: Visual-spatial processing; Conceptual processing
Conversing: MLR5 Co-Craft Questions.
Display the initial task statement that begins, “Imagine there are five cards...", before revealing the questions that follow. Ask pairs to write down possible mathematical questions that can be answered about the situation. As pairs share their questions with the class, listen for and highlight questions that ask how the “without putting the card back” part of the scenario would change the outcome of the sample space. This will help students consider the differences between this problem and previous problems and how that impacts the sample space.
Design Principle(s): Optimize output; Maximize meta-awareness
Student Facing
Imagine there are 5 cards. They are colored red, yellow, green, white, and black. You mix up the cards and select one of them without looking. Then, without putting that card back, you mix up the remaining cards and select another one.
Write the sample space and tell how many possible outcomes there are.
What structure did you use to write all of the outcomes (list, table, tree, something else)? Explain why you chose that structure.
What is the probability that:
You get a white card and a red card (in either order)?
You get a black card (either time)?
You do not get a black card (either time)?
You get a blue card?
You get 2 cards of the same color?
You get 2 cards of different colors?
Student Response
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Student Facing
Are you ready for more?
In a game using five cards numbered 1, 2, 3, 4, and 5, you take two cards and add the values together. If the sum is 8, you win. Would you rather pick a card and put it back before picking the second card, or keep the card in your hand while you pick the second card? Explain your reasoning.
Student Response
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Anticipated Misconceptions
Students may misread the problem and think that they replace the card before picking the next one. Ask these students to read the problem more carefully and ask the student, “What is possible to get when you draw the second card while you already have a red card in your hand?”
Activity Synthesis
The purpose of the discussion is for students to compare the same context with replacement and without replacement.
Consider asking these questions for discussion:
“What would change about your calculations if the experiment required replacing the first card before picking a second card?” (There would be 25 outcomes in the sample space. The probability of getting the same color twice would be
\(\frac{5}{25}\)
. The probability of getting different colors would be
\(\frac{20}{25}\)
. The probability of getting red and white would be
\(\frac{2}{25}\)
. The probability of getting a black card would be
\(\frac{9}{25}\)
and not getting a black card would be
\(\frac{16}{25}\)
. It would still be impossible to get a blue card, so its probability would be 0.)
“What do you notice about the sum of the probability of getting a black card and the probability of not getting a black card?” (They have a sum of 1.)
“Explain why these outcomes might have probabilities with this relationship.” (Since you either get a black card or not, together their probabilities should be 1 or 100%.)


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