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.

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.

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

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.

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 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.

Functions: students learn to define, use, and evaluate functions to model relationships between quantities.
problem_activity: Warm-up
The purpose of this warm-up is to elicit the idea that there are different ways to collect and represent data. This will be useful when students examine statistical designs in a later activity. While students may notice and wonder many things about these images, how the data are represented (or misrepresented) is the important discussion point.
Through articulating things they notice and things they wonder about the headlines and graph, students have an opportunity to attend to precision in the language they use to describe what they see (MP6). They might first propose less formal or imprecise language, and then restate their observation with more precise language in order to communicate more clearly.
Launch
Display the image and headlines for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice and wonder with their partner, followed by a whole-class discussion.
Student Facing
A graph and 2 headlines from a website are shown.
What do you notice? What do you wonder?
[IMG]
“80% of Dentists Recommend Acme Toothpaste”
“Pythagoras Brand Rulers Measure 20% Better”
Student Response
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Activity Synthesis
Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the image or statement. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information.
Display a wider view of the graph that shows more of the story.
[IMG]
Ask, “How does having the whole graph tell a different story than just having the part of the graph from 1998 to 2012?” (In the whole graph, you can see that the slope of the line is positive over the 112 years represented in the graph, and you could find other 12-year periods in which the slope is negative, positive, or close to zero.) If time allows, follow up by asking, “Does the dashed line appear to be a line of best fit or is it something else? Explain your reasoning.” (The dashed line does not appear to be a line of best fit. It just seems to connect the first point in the graph to the last point. This could be misleading since the line of best fit would have a greater slope than the dashed line, and would be a little lower on the graph.)
Here are some questions for discussion:
“Why is the dentist headline potentially misleading?” (It does not mention if they recommend other brands as well. For example, dentists might just recommend using any toothpaste at all and this brand works just as well as any other brand.)
“Why is the ruler headline misleading?” (It does not mention what it is that Pythagoras rulers measure better than.)


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