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.

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.

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.

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

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.

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

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

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

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.

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

Number & Operations in Base Ten: students learn to work with the base-ten system and build place value understanding.
problem_activity: Activity
In this activity, students draw a bar graph and histogram, then describe the distributions shown on each display. Although the two visual displays may appear similar at first glance, there are important distinctions between the representations. Students notice differences in how we might characterize distributions in bar graphs and those in histograms, including how we describe typical values or categories. Along the way, students consolidate their understanding about categorical and numerical data.
Launch
Students will need the data on their travel methods and times, collected at the beginning of the unit. Distribute or display the data collected for these questions from the survey given earlier in the unit. Alternatively, complete the tables in the blackline master ahead of time.
Arrange students in groups of 2. Give one copy of the blackline master to each group of students. Display the data from the prior survey or the completed frequency tables for all to see or give a copy to each group of 2 students. Give students 5–6 minutes to complete the activity. Ask one partner to create a bar graph to represent the data on the class’s travel methods and the other to create a histogram to represent the data on travel times, and then answer the questions together.
Student Facing
Your teacher will provide you with some data that your class collected the other day.
Use the data to draw a histogram that shows your class’s travel times.
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Describe the distribution of travel times. Comment on the center and spread of the data, as well as the shape and features.
Use the data on methods of travel to draw a bar graph. Include labels for the horizontal axis.
[IMG]
Describe what you learned about your class’s methods of transportation to school. Comment on any patterns you noticed.
Compare the histogram and the bar graph that you drew. How are they the same? How are they different?
Student Response
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Student Facing
Are you ready for more?
Use one of these suggestions (or make up your own). Research data to create a histogram. Then, describe the distribution.
Heights of 30 athletes from multiple sports
Heights of 30 athletes from the same sport
High temperatures for each day of the last month in a city you would like to visit
Prices for all the menu items at a local restaurant
Student Response
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for free access to Extension Student Response.
Activity Synthesis
The purpose of the discussion is for students to recognize the differences between histograms and bar graphs.
Select one student to show a completed histogram and another to show a completed bar graph. Then, solicit several observations about how the two graphical displays compare. Ask questions such as:
“How are the bar graphs and histograms alike? How are they different?”
“Can we use a bar graph to display the data on travel times? Why or why not?”
“Can we use a histogram to display the data on methods of travel? Why or why not?”
Next, select a few other students to share their descriptions of the distributions shown on each type of display. Then, ask questions such as:
“How are your descriptions of the distribution for travel methods different than those for travel times?”
“Can you talk about the shape of a distribution shown on a bar graph? Can you talk about its center and spread? Why or why not?”
Students should recognize that only the distribution of numerical data can be described in terms of shape, center, or spread. We cannot analyze these features for a distribution of a categorical data on a bar graph because a bar graph does
not
use a number line. This means the bars can be drawn anywhere, in any order, and with any kind of spacing, so shape, center, and spread would have no meaning.
Action and Expression: Internalize Executive Functions.
Provide students with a graphic organizer to record differences between histograms and bar graphs.  Consider creating a shared display and keeping it visible throughout the unit.
Supports accessibility for: Language; Organization
Representing, Conversing: MLR2 Collect and Display.
As students discuss the similarities and differences between histograms and bar graphs, record and display phrases you hear in a chart, such as a Venn diagram. Use this as a reference when highlighting or revoicing academic vocabulary from this unit (frequency, bins, numerical data, categorical data). This will help students connect unique and shared characteristics of histograms and bar graphs while reinforcing mathematical language.
Design Principle(s): Maximize meta-awareness


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