Lesson 8

The purpose of this warm-up is for students to estimate heights and discuss their estimates using the language of inequalities.As students discuss their estimates with a partner, monitor the discussions and identify students who use different estimation strategies to share during the whole-class discussion.

Arrange students in groups of 2. Give students 1 minute of quiet work time to complete the first question. Pause after the first question and ask students to share their responses. Record and display them for all to see. Tell students the height of the man in the pictures is 5 feet 10 inches and give them 1 minute to discuss the second question with their partner. Follow with a whole-class discussion focused on the second question.

During the whole-class discussion, encourage students to use language such as “his height is more than . . . ” and “her height is less than . . . ” instead of “taller than . . .” and “shorter than . . . ” so they become familiar with the more precise language of inequalities.

Ask students to share their responses to the second question. As students share, ask them how to write their explanation as a sentence and inequality. For example, if students say, “The child is shorter than 5'10",” ask them how we could write that statement using “greater” or “less” and using an inequality. Record and display their responses for all to see. If students do not say, “The child’s height is less than 5'10" and \(h<5\text '10\text "\),” where \(h\) represents the child’s height, then make these statements explicit.

If there is time, ask students for the difference between their estimates and the actual heights.

In this activity, students extend their understanding of inequality statements by considering an unknown quantity with one or more constraints in a real-world situation. Students represent situations with inequality statements and reason about possible values that make them true, showing the possibilities on a number line. They realize that these situations often have many values that make the inequality true, and often even an infinite number, making the number line representation of possible values very useful (MP7). Students examine the difference between solutions that are continuous (can take on any value, usually involving measurement of an amount) and those that are discrete (require whole number solutions because of the context, usually involving a count of people or objects) (MP2). They also consider when to include or exclude the endpoints. They learn how to represent this on the number line with a closed (include) or open (exclude) circle at the boundary of the constraint. Note that these materials don’t introduce \(\geq\) or \(\leq\) symbols until grade 7, but some individual teachers prefer to introduce them earlier.

Here are some topics for discussion:

• Differences between solutions to the people problem and the weight problem (continuous v. discrete).
• Inclusion and exclusion of 9 and how to distinguish them on the number line (including open and closed circle).
• The idea that there are many values that make these inequalities true. At times, there are an infinite number of values that make the inequality true.
• Advantages of using inequality statements or graphs instead of listing or describing the solutions.

There are some additional restrictions in some of the scenarios. For example, when counting people in a boat or measuring weight, zero is the least value that makes sense. For the people at the magic show and the amount of butter, very large numbers satisfy the inequalities; however, they don’t necessarily make sense in the contexts presented. These restrictions are not a focus for this activity, but they are likely to come up during the discussion.

Toward the end of the discussion, display the statement “Jada built a robot that pushes small boxes from one place to another. The robot is able to push up to 3 pounds. For what box weights (\(w\)) can the robot push the box?” Ask students to represent all possible answers in three ways:

1. By listing or describing all numbers that answer the question. (Any weight that is equal to or less than 3 pounds.)
2. By plotting or graphing the values that work on a number line. (Closed circle at 3 and shaded to the left.)
3. By writing an inequality. (Using the given variable) (\(w<3\) and \(w=3\).)

The purpose of this task is to extend the use of inequalities to describe maximum and minimum possible values, using symbols and determining whether or not a particular value makes an inequality true. Though students are thinking about whether or not a particular value makes an inequality true, the term “solution” will not be formally introduced until a future lesson.

Students will estimate the maximum and minimum height of a basketball hoop in a given picture. They will then explore how inequalities with these 2 values, the maximum and minimum, can describe the restrictions and possible heights for the hoop. The inequalities are represented as symbolic statements as well as on number lines.

Once they have created inequalities, students will decide if a given value could be a possible height for the hoop. For the last question, students work with a partner so each will have a chance to check a possible solution to an inequality that they didn’t estimate and write themselves.

Arrange students in groups of 2. Give students 8 minutes of quiet work time to complete questions 1 through 4 followed by 2 minutes partner discussion for question 5. Follow with whole-class discussion.

Students may understand that maximum means the largest value but may have trouble with the concept that all other values are therefore less than the maximum. A similar misunderstanding may occur with the concept of minimum value. For these students, have a short conversation about their maximum and minimum height estimates and how those values restrict the possible heights of the hoop.

The discussion should focus on the concepts of minimum and maximum. While students are not yet using the formal definition of “solutions to an inequality,” they are reasoning about whether a value makes an inequality statement true. A student’s estimated minimum is the least value that makes their inequality true. Similarly, their maximum is the greatest value that makes their inequality true.

Ask students to share their responses to question 5. Select one group to share an estimate that makes a particular inequality true and an estimate that makes the same inequality false. Ask students to describe the values that make both inequalities true at the same time. Record their responses and display for all to see. Include both inequality symbols and number line representations.