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.

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.

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

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.

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

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

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.

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.
problem_activity: Warm-up
The purpose of this warm-up is for students to begin to see the need for samples of data when the population is too large. In this activity, students are asked to think about a question involving all the students at their school and compare the question to an earlier lesson in which the population was small and it was easy to obtain data for the entire population.
Teacher Notes for IM 6–8 Accelerated
The activity referring to song lengths in the second question is not included in IM 6–8 Math Accelerated. If necessary, tell students that the albums each contained either 7 or 8 songs.
Launch
Give students 2 minutes of quiet work time followed by a whole-class discussion.
Student Facing
Consider the question: In general, do the students at this school have more letters in their first name or last name? How many more letters?
What are some ways you might get some data to answer the question?
The other day, we compared the heights of people on different teams and the lengths of songs on different albums. What makes this question about first and last names harder to answer than those questions?
Student Response
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Activity Synthesis
The purpose of the discussion is to highlight the methods of getting data for the school more than it is the method of computing the answer.
Select some students to share their responses.
Students who have elected to sum all the letters in the first names in the school and all the letters in the last names in the school may note that it is a simple comparison to tell whether there are more in first or last names, since you get one single large number for each group. (Comparing data sets.)
Students who have elected to calculate the mean for each group and use MAD as a method of comparison may note that while the calculations may take more time, they give you more precise information, such as knowing about how long first names and last names are, as well as a way to compare the two sets. (Using the general rule from the previous lesson.)
Students who suggest surveying a small group of students may point out that it would be easier to do the calculation with a smaller group.  The information would not be as accurate, but it would take a lot less time and might give a good general idea. It would depend on how accurate you needed your answer to be.  (Introduction to sampling.)


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