In this activity, students extend the use of inequalities to describe maximum and minimum possible values. 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 estimate the maximum and minimum height of a basketball hoop in a given picture, represent these estimates on a number line, and use inequality symbols to write statements. Note that the symbols ≤ and ≥ will be introduced in a later course and are not used in this unit. To describe a range of values that includes the boundary value, such as “ is 6 or greater,” two statements are used: and .

The purpose of this discussion is to introduce graphing inequalities on a number line and to discuss boundary values. Begin by inviting students to share the inequality statements they wrote to describe the minimum height of the basketball hoop, recording their responses for all to see. Then display a blank number line for all to see.

Choose a student’s inequality statement to use as an example, such as , and demonstrate how to represent this on the number line by labeling a tick mark as 6, drawing an open circle at 6, and shading the area to the right of the circle. Label the shaded part of the number line with the statement .

Then ask students what the word “minimum” means to them. (the smallest value that something can be) Ask students:

• “If the minimum height of the basketball hoop is 6 feet, could the height be equal to 6 feet?” (Yes, the minimum refers to the smallest value the height could be equal to.) 
• “How do we use numbers and symbols to write that the height could be equal to 6 feet?” ()

Remind students that to represent on the number line, we would plot a point at 6. Adding this to the number line would make it look like this:

Repeat the explanation, and draw another number line to represent the maximum height of the basketball hoop. Make sure students understand that is graphed to the left of 15 and does not include the value 15. But the term “maximum” implies inclusion of 15, so the statement can also be graphed on the same number line as a point at 15.

Students sort different number lines, descriptions of real-world situations, and descriptions of possible numerical values during this activity. A sorting task gives students opportunities to analyze representations, statements, and structures closely and make connections (MP7).

Students 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 (MP2). 

The goal of this discussion is for students to consider how the context of a situation can affect different aspects of the solution, such as whether the solution is continuous or discrete, whether negative values make sense, and how the boundaries behave. Discuss the following questions:

• “Why did some stories have number lines with just dots?” (Those stories were referring to numbers of people. Since there can’t be a fraction of a person, it makes sense to represent those situations with only whole numbers.)
• “Why did the graph representing the food scale stop at 0 instead of having an arrow pointing to the left?” (It doesn’t make sense to measure something with a negative weight, so negative numbers don’t make sense in this situation.)
• “Why did the graph representing the butter Lin needed for cookies have an unshaded, or open, circle at 9?” (Lin needed more than 9 ounces of butter. Since exactly 9 ounces of butter would not have been enough, we leave the circle at 9 open, or unshaded.)
• “Why does the graph representing the butter Lin needs have an arrow pointing to the right? Is this reasonable?” (There is an arrow because Lin needs more than 9 ounces, but we don’t know the maximum amount, so any amount greater than 9 would work. But realistically, values greater than 9 that are very large are not reasonable.)

If time allows, display this statement for all to see:
“Jada built a robot that can push heavy boxes from one place to another. The robot is meant to be used only for pushing boxes heavier than 100 pounds. For what box weights () should the robot push the box?” 

Ask students to represent all possible answers in three ways:

1. List or describe all numbers that answer the question. (any weight that is greater than 100 pounds)
2. Plot or graph the values that work on a number line. (open circle at 100 and shaded to the right)
3. Write an inequality. ()