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

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

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

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

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

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.

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.

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

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.

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.

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.
problem_activity: Activity
The mathematical purpose of this activity is for students to begin to understand the relationship between sample size and margin of error. Two characters collect samples and notice that their simulations result in very different margins of error. Students should begin to notice that a larger sample size should result in an estimate with a lesser margin of error.
Launch
Reading: MLR6 Three Reads.
Use this routine to support reading comprehension of this problem. Ask students to keep their books or devices closed and display only the task statement without revealing the questions that follow. Use the first read to orient students to the situation. After a shared reading, ask, “What is this situation about?” (two students found a proportion from a random sample). After the second read, students list any quantities that can be counted or measured, without focusing on specific values (proportions, margins of error, and number of simulations). During the third read, the question or prompt is revealed. Invite students to discuss possible strategies to answer the questions, referencing the relevant quantities named after the second read. This helps students connect the language and reasoning needed to solve a problem while maintaining the cognitive demand of the task.
Design Principle: Support sense-making
Representation: Internalize Comprehension.
Begin by providing students with questions to help them form generalizations about mean, sample size, standard deviation, and margin of error. For example, “When the mean changes, does the margin of error change? When the standard deviation changes, does the margin of error change?”
Supports accessibility for: Conceptual processing
Student Facing
Elena and Clare are each working on a project about how high school students are having trouble finding jobs. They each find the proportion of students without jobs from a random sample, then use a computer to do 1,000 simulations using the proportion they found and report the results.
Elena says, “The proportion of high school students without jobs is about 0.70 with a margin of error of 0.280.”
Clare says, “The proportion of high school students without jobs is about 0.74 with a margin of error of 0.138.”
Both students reported the margin of error based on 2 standard deviations from their simulations. What are the mean and standard deviation each student found? For at least one student, show your reasoning.
Clare and Elena try to figure out why Clare had such a smaller range of values in her report.
First they consider the proportion they used in the simulations. Elena says, “My simulation used 0.7 as the proportion since I found that proportion in my original sample.” Clare says, “My simulation used 0.75 as the proportion since I found that proportion in my original sample.” The students used different proportions in their simulations. Do you think this is why Clare has a smaller margin of error? Explain your reasoning.
They look for more differences in their initial sample and discover than Elena surveyed 10 people in her initial sample and Clare surveyed 40 people. Do you think this is why Clare has a smaller margin of error? Explain your reasoning.
Student Response
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Activity Synthesis
The purpose of this discussion is for students to think about the relationship between sample size and margin of error. Ask students,
“Why do you think Clare continues to get a smaller margin of error than Elena?” (She took a bigger sample so the samples are more likely to be representative of the population.)
“Elena had a sample size of 10 and Clare had a sample size of 40. Why do you think the sample size could have caused Clare’s margin of error to be less than Elena’s margin of error?” (Since there are more people included in each sample, the samples themselves capture more of the population variability, so each sample estimate is more likely to accurately reflect the population characteristic.)


correct_option_index: 9