Lesson 1

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 an 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 are, as a class. Is the actual value inside their range of values? Is it towards the middle? How variable are their estimates? What are 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 create a two-way table and practice using it to interpret data. In the associated Algebra 1 lesson, students interpret two-way tables, and they get to learn or review how to create a two-way table through movement. 

Arrange the students’ seats in rows and place three labels at the front of the classroom that are side by side. The labels represent the categories for the electronic device that students prefer: laptops, tablets, and other. Ask students to stand on the left, center, or right side of the classroom based on their preference for the type of device they like to use to do classwork. Add 2 more categories, basketball and baseball, to the left side of the room. Students stay on the left, center, or right side of the room as they are already arranged, and move forward or backward to represent their preferences for the second set of categories. Each student should be standing in one of four possible groups. Display the numbers, creating a two-way table.

Select a particular student and ask the class what information they know about the student based on where they are standing in the room.

The goal of this discussion is for students to think about how to construct and interpret data. Here is a sample table displaying made up data for a class. 

	laptops	tablets	other	total
basketball	4	8	2	14
baseball	2	9	1	12
total	6	17	3	26

Discuss students’ understanding of using a table to interpret data and ask questions about the data points that are not addressed in the questions. Here are sample questions to promote a class discussion:

• “What is the difference between a table and a two-way table?” (Two-way tables are a specific type of table that can be used to organize data into two categories at once.)
• “What strategies do you use to answer questions about specific categories?” (I was sure to circle the entire column or row that the question asked about so that I could narrow down the focus on the table and have a more clear idea of what was being asked of me.)
• “What does the number 6 mean in the table?” (6 is the total number of students who prefer laptops.)
• “What does the number in the bottom right of the table represent?” (It is the total number of students who were surveyed in my class.)
• “How many students prefer baseball?” (12)
• “How many students prefer baseball and tablets?” (9)
• “How many students prefer basketball and laptops?” (4)

The purpose of this activity is for students to practice interpreting data from a two-way table. This prepares students to analyze and interpret data in the associated Algebra 1 lesson.

Allow students to complete the activity individually.

Discuss how students interpret the table. Here are questions to promote class discussion:

• “How can you tell which number represents the total number of students who were surveyed?” (I look for the portion of the table with the total that includes the numbers from all of the categories.)
• “What do you find challenging about interpreting a two-way table?” (Having multiple categories can make the data difficult to interpret.)
• “What are some advantages of organizing data with a two-way table?”(It can make categorical data easier to organize, sort, and interpret.)