The purpose of this Warm-up is to practice the skill of estimating a reasonable answer based on experience and known information, and also to help students develop a deeper understanding of the meaning of standard units of measure. It gives students a low-stakes opportunity to share a mathematical claim and the thinking behind it (MP3). Asking, “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).

The goal of this discussion is for students to analyze the reasonableness and accuracy of their estimates.

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 responses 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 ? Is anyone’s estimate greater than ?” If time allows, ask students, “After this discussion, does anyone want to revise their estimate?”

Then reveal the actual value and add it to the display.

Encourage students to think about and discuss the accuracy of their estimates and the estimates that are displayed by asking questions such as:

• "Is the actual value inside the range of values?"
• "Is it toward the middle?"
• "How variable are the estimates?"
• "What are the sources of error?"

Consider developing a method to record a "snapshot" of the estimates and the actual value so that students can track, over time, their progress as estimators.

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. As students find their place in the two-way table and make sense of the different cells, they make sense of problems and persevere in solving them (MP1).

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 did 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 the 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. Students reason abstractly and quantitatively (MP2) when they interpret the meaning of the numbers in the two-way table.

The goal of this discussion is for students to explain how they completed and interpreted the two-way table.

Direct students’ attention to the reference created using Collect and Display. Ask students to share how they filled in the table. Invite students to borrow language from the display as needed. As they respond, update the reference to include additional phrases. For example, the display may have “I filled in the boxes” already on it and can be updated with the phrase “I added the values in the cells in a row or column to calculate the total.”

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 part 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.)