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 measurement. 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 estimates 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 estimates 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, over time, track their progress as estimators.

The purpose of this activity is to give students an opportunity to practice checking whether a particular value is a solution to an equation, and to recall properties of operations and equality that preserve the solution set of an equation.

This will be useful, in the associated Algebra 1 lesson, when students consider when and why equations have the same solution. Developing multiple approaches to determining if a value is a solution allows students to reason abstractly and quantitatively (MP2).

Monitor for students who use these different approaches:

• Check the value of in only one expression and reason about the second expression

• Identify properties by name, such as “commutative,” “distributive,” and “associative”

• Identify properties informally, such as “changing the order doesn’t matter” or “you can add first and then multiply, or multiply first and then add” or “you can multiply three or more terms in any order” or “you can add the same thing to each side”

• Manipulate the expressions

• Graph one or both expressions

Making graphing technology available gives students an opportunity to choose appropriate tools strategically (MP5).

The purpose of this activity is for students to practice generating equivalent equations. The structure of taking turns and justifying your thinking to your partner supports students to check their work as they go, and the class has a chance to check for equivalence when the total score is recorded.

Students also have to justify, whether formally or informally, why each move keeps the equation equivalent. This will be helpful for students, in an associated Algebra 1 lesson, when they must determine what moves are acceptable to make to an equation without changing its solution. As students justify the equivalence of the equations, they are reasoning abstractly and quantitatively (MP2).

Monitor for students who solve the equation first and then work to create new equations with the same answer, and for students who simply create new equivalent equations from the given equation. Monitor for students who add or multiply by the same number on each side of an equation and students who use properties of operations (distributive, commutative, associative).