Lesson 3

The purpose of an Estimation warm-up is to practice the skill of estimating a reasonable answer based on experience and known information. It gives students a low-stakes opportunity to share a mathematical claim and the thinking behind it (MP3). Asking yourself “Does this make sense?” is a component of making sense of problems (MP1), and making an estimate or a range of reasonable answers with incomplete information is a part of modeling with mathematics (MP4).

Display the image for all to see. Ask students to silently think of a number they are sure is too low, a number they are sure is too high, and a number that is about right, and write these down. Then, write a short explanation for the reasoning behind their estimate.

Ask a few students to share their estimate and their reasoning. If a student is reluctant to commit to an estimate, ask for a range of values. Display these for all to see in an ordered list or on a number line. Add the least and greatest estimate to the display by asking, “Is anyone’s estimate less than \(\underline{\hspace{.5in}}\)? Is anyone’s estimate greater than \(\underline{\hspace{.5in}}\)?” If time allows, ask students, “Based on this discussion does anyone want to revise their estimate?” Then, reveal the actual value and add it to the display. Ask students how accurate their estimates were, as a class. Was the actual value inside their range of values? Was it towards the middle? How variable were their estimates? What were the sources of the error? Consider developing a method to record a snapshot of the estimates and actual value so that students can track their progress as estimators over time.

The purpose of this activity is for students to recall how to read a relative frequency table and to determine if variables have an association. This prepares students for work in the associated Algebra 1 lesson when they create and interpret relative frequency tables.

Students should have an understanding of what STEM careers are. STEM careers include careers in the fields of science, technology, engineering, math, and medicine. Ask students, “What are some examples of careers in STEM?” Responses include: doctors, computer scientists, civil engineers, computer scientists, and architects. 

Discuss how students interpreted the tables and how they decided if there is an association. Remind students that a decimal for the number of students is unrealistic, since there cannot be a fraction of a person. It is approporiate for students to round down to the nearest whole number for this reason. Here is a question for class discussion:

• "How do you determine if there is an association between the two variables?"(Find the percentage of each category in a row or in a column. Compare the percentages for the categories to see if they are similar. If the percentages are similar, there is not likely an association. If they are very different, there is a possible association)

The purpose of this activity is for students to practice creating pairs of variables that they think are associated. It is not important for students to focus on the type of association they think the variables may have. This preprares students for work in the associated Algebra 1 lesson when they create pairs of variables that are and are not associated, and create data to represent each. 

Display for all to see examples of variables that have an association and ones that do not. Allow a whole-class discussion about how each pair of variables might be associated, and why it makes sense for some variables to have no association. 

Variables with an association:

• season and whether half or more of the people outside are wearing jackets or not
• growth in plant length and whether someone waters their plants

Variable with no association:

• mobile device preference and eye color
• taking a visual art elective or not and mode of transportation to school 

Discuss students' understandings of associations. Here are sample questions for class discussion:

• "How did you determine if two variables are associated?"(I consider if the two things have any type of relationship. I also think about what would happen to one variable if the other changes at all.)
• “Can you think of two variables that might seem unrelated at first, but might actually be associated? Explain your reasoning.” (Shark attacks and whether a person has eaten ice cream recently. Although they seem unrelated, both things tend to increase in the summer and decrease in the winter.)