The purpose of this Warm-up is to elicit students’ current understanding of the inequality symbols, which will be useful when students use inequalities to represent situations later in this lesson. In addition, the activity is designed to elicit students’ understanding and possible misconceptions about inequalities involving negative numbers. Students reason abstractly and quantitatively (MP2) as they compare each pair of numbers.

The purpose of this discussion is to understand what students know about inequality symbols. Ask students to read an inequality statement out loud. If students struggle to read the statements accurately, demonstrate how to say “five is less than ten” and “one-fifth is greater than one-tenth.” Here are some questions for discussion:

• “How do you know which symbol to use when both numbers are positive?” (I compare the numbers and see which number is farther from 0. That number is greater and the other is less.)
• “How does having one positive number and one negative number change your approach?” (Negative numbers are always less than positive numbers. When there is one of each, I do not have to think as much about the numbers, just the signs.)
• “How does having two negative numbers change your approach?” (Since they have the same sign, I have to compare the numbers. For negative numbers, being closer to 0 is greater than being farther from 0.)

The mathematical purpose of this activity is to provide a context for students to reason about inequalities and a range of values that can work for a given situation. In addition, this activity provides a context for reasoning about inequalities involving negative numbers. Students construct viable arguments (MP3) when connecting inequalities with graphs and explaining which gear would be best based on the temperature.

This activity builds on students' prior work with negative numbers (in grade 6), in which contexts and location on the number line were used to support students’ understanding of ordering negative numbers.

This work will be helpful when students solve inequalities and interpret the solutions in their Algebra 1 class, especially when working with inequalities involving negative numbers.

The goal of this discussion is to help students make sense of the inequality statements given in the problem. Begin by asking students which gear they matched to each inequality.

Display the three inequalities for all to see, labeled with which piece of gear they go with:

1. Down jacket:  and 
2. Down sleeping bag: 
3. Synthetic sleeping bag: 

Call on different students to read each inequality out loud. If students read the inequalities as: “Negative twenty is less than or equal to , and is less than or equal to 20,” confirm the correctness of their reading. Then ask them to translate that into a statement that is easier for customers to make sense of, such as, “The down jacket is suitable for temperatures between -20 and 20.”

Discuss how students matched the inequalities with graphs. Here are some questions for discussion:

• "How did you decide which graph matches the down jacket?" (It is the only description that has two constraints, and its graph is the only graph to show stopping points at two different values.)
• "On the graph of an inequality, what is the difference between an open circle and a closed circle?" (An open circle means the value is not included in the solution, and a closed circle means that the number is included in the solution.)
• "What is the difference between the meanings of Graph B and Graph E?" (Graph E has a closed circle, meaning the sleeping bag is comfortable at or above . Graph B has an open circle, meaning the sleeping bag is comfortable at any temperature that is above .)

The purpose of this activity is to practice representing situations with inequalities. This will be helpful for students in the associated Algebra 1 lesson when they represent constraints of a more complex situation. Students reason abstractly and quantitatively (MP2) as they attempt to symbolically represent a situation.

The goal of this discussion is to connect verbal descriptions with symbolic representations of inequalities. Display these inequalities for all to see. Select several students to read each of these inequalities. Encourage them to read what the statement means, not just decode each symbol—for example, reading “ is between 3 and 7” rather than “3 is less than or equal to " and " is less than or equal to 7.”

•  and 

After students read the second and third inequalities, ask, “What is another way to say the same relationship?” If not brought up in students’ explanations, discuss how “-5 is greater than ” is equivalent to “ is less than -5.”

Give students 1 minute of quiet think time to think about answers to these questions before discussing their answers as a class:

• “What do you remember learning about inequalities before today?” (I remembered the meaning of the symbols, and that inequalities can also include  and .)

• “What did you get reminded about as a result of today's lesson?” (I was reminded that it can be confusing to think about values in relation to a negative number.)