standard_description: Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., p^(2)). For example, by truncating the decimal expansion of Ö2, show that Ö2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.
problem_activity: Warm-up
The purpose of this warm-up is to elicit the idea that data can appear in clusters, which will be useful when students work with a variety of scatter plots in the future. While students may notice and wonder many things about these graphs, how the data are clustered is the important discussion point.
When students articulate what they notice and wonder, they 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
Keep students in groups of 2. Allow 2 minutes quiet think time followed by partner and whole-class discussion.
Display the scatter plots 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
What do you Notice? What do you Wonder?
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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 scatter plot. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information.
If the clustering of the data in the scatter plots does not come up during the conversation, ask students to discuss this idea. Tell students that when data seems to have more than one pattern, it is called clustering. Clustering of the data like in graph B, C, and D can reveal hidden patterns. Usually, clustering means there are subgroups within our data that may represent different trends.
For example, in Plot B, the data may represent body measurements of a certain species of bird. Although the data originally came from a group that made sense (a single species), there appear to be subgroups that have a large influence on the data as well. The lower half of the data may represent females and the upper half may represent males, so we can see that there are different patterns within the different subgroups.
When clustering is present, it may be helpful to investigate the cause of the separation and analyze the data within the subgroups rather than as a whole.


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