In this activity, students critique a statement or response that is intentionally incorrect (MP3). The example given represents a common mistake made by students and prompts students to consider the meaning and placement of inequality symbols in relation to a number line.
Launch
Give students 2–3 minutes of quiet work time, and follow with a whole-class discussion.
Activity
None
Andre drew this number line to represent .
Do you agree with Andre’s number line? Explain your reasoning.
Student Response
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Building on Student Thinking
Activity Synthesis
The purpose of this discussion is to address a common mistake made by students when graphing or interpreting an inequality where the variable is on the right side of the inequality. Begin by inviting 2–3 students to share their responses and reasoning. If not mentioned by students, discuss the following questions:
“How do we say the inequality ?” (15 is less than n.)
“What are some values that could be?” (16, 20, 34.9, 983)
“What are some values that could not be?” (15, 14.9, 10, 1, -3)
“Since could not be equal to 15, did Andre draw the correct type of circle at 15?” (Yes, it should be an open, non-shaded circle.)
“How could Andre’s number line be changed to correctly represent ?” (The numbers to the right of 15 should be shaded instead of to the left.)
13.2
Activity
Standards Alignment
Building On
Addressing
6.EE.B.5
Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.
Write an inequality of the form or to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form or have infinitely many solutions; represent solutions of such inequalities on number line diagrams.
In this activity, students are formally introduced to the phrase “solution to an inequality” as a number that can be used in place of the variable to make the inequality true. Students use given height restrictions for a variety of amusement park rides to explore these ideas. They represent the height restrictions as inequality statements and graph those inequalities on the number line. Students reason abstractly when determining whether a value is a solution to one or more of the inequalities and what that means in context (MP2).
This activity uses the Stronger and Clearer Each Time math language routine to advance writing, speaking, and listening as students refine mathematical language and ideas.
Adjust the time for this activity to 15 minutes by skipping the Stronger and Clearer Each Time routine in the Activity Synthesis.
Launch
Arrange students in groups of 2. Display the inequality for all to see. Ask students to determine a value for that will make the inequality true, and record student responses for all to see.
Tell students that just as a solution to an equation is a value of the variable that makes the equation true, a solution to an inequality is a value of the variable that makes the inequality true. And while equations generally have one solution, inequalities have many, sometimes infinitely many, solutions.
Give students 7–8 minutes of quiet work time followed by 1–2 minutes for a partner discussion. Follow with a whole-class discussion.
Engagement: Provide Access by Recruiting Interest. Invite students to generate a list of additional examples of inequalities that connect to their personal backgrounds and interests. Supports accessibility for: Conceptual Processing, Attention
Activity
None
Priya finds these height requirements for some of the rides at an amusement park.
To ride the . . .
you must be . . .
High Bounce
between 55 and 72 inches tall
Climb-A-Thon
under 60 inches tall
Twirl-O-Coaster
58 inches minimum
Write equations and/or inequalities for the height requirements of each ride. Use for the unknown height. Then, represent each height requirement on a number line.
High Bounce
Climb-A-Thon
Twirl-O-Coaster
Han’s cousin is 55 inches tall. Han doesn’t think she is tall enough to ride the High Bounce, but Kiran believes that she is tall enough. Do you agree with Han or Kiran? Be prepared to explain your reasoning.
Priya can ride the Climb-A-Thon, but she cannot ride the High Bounce or the Twirl-O-Coaster. Which of the following could be Priya’s height? Be prepared to explain your reasoning.
59 inches
53 inches
56 inches
Jada is 56 inches tall. Which rides can she go on?
Kiran is 60 inches tall. Which rides can he go on?
The inequalities and represent the height restrictions, in inches, of another ride. Write three values that are solutions to both of these inequalities.
Student Response
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Building on Student Thinking
Activity Synthesis
The purpose of this discussion is for students to communicate their understanding of what it means to be a solution to an inequality. Display the question for all to see, and give students 2–3 minutes to draft a response:
“What does it mean to be a solution to an inequality?”
Use Stronger and Clearer Each Time to give students an opportunity to revise and refine their response. In this structured pairing strategy, students bring their first draft response into conversations with 2–3 different partners. They take turns being the speaker and the listener. As the speaker, students share their initial ideas and read their first draft. As the listener, students ask questions and give feedback that will help their partner clarify and strengthen their ideas and writing.
If time allows, display these prompts for feedback:
“ makes sense, but what do you mean when you say . . . ?”
“Can you describe that another way?”
“How do you know . . . ? What else do you know is true?”
Close the partner conversations, and give students 3–5 minutes to revise their first draft. Encourage students to incorporate any good ideas and words they got from their partners to make their next draft stronger and clearer. Here is an example of a second draft:
“When a number is a solution to an inequality, that number could be used in place of the variable and the statement would be true.”
If time allows, invite students to compare their first and final drafts. Select 2–3 students to share how their drafts changed and why they made the changes they did.
13.3
Activity
Standards Alignment
Building On
Addressing
Building Toward
7.EE.B.4.b
Solve word problems leading to inequalities of the form or , where , , and are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem. \$
In this activity, students express an inequality that includes the boundary value using only the symbols and . This generates a need for the symbol , which is introduced in the Activity Synthesis. Students also recall the fact that a closed circle is used to graph an inequality that includes the boundary value.
Monitor for students who express the answer to the last question using words or using symbols. The responses to the last question will be used to introduce the new notation.
Students attend to precision (MP6) when they carefully choose the symbols to express the possible values for the height in the last question.
This activity uses the Collect and Display math language routine to advance conversing and reading as students clarify, build on, or make connections to mathematical language.
Launch
Arrange students in groups of 2. Allow students 5–10 minutes quiet work time and time to share their responses with a partner, followed by a whole-class discussion.
Use Collect and Display to create a shared reference that captures students’ developing mathematical language. Collect the language that students use to express the possible values for Noah’s friend’s height. Display words and phrases, such as “at least,” “greater than,” and “greater than or equal to.”
Representation: Access for Perception. Read all problems aloud. Students who both listen to and read the information will benefit from extra processing time. Supports accessibility for: Language, Attention
Activity
None
A sign next to a roller coaster says, “You must be at least 60 inches tall to ride.” Noah is happy to know that he is tall enough to ride.
Noah is inches tall. Which of the following can be true? Explain how you know.
Noah’s friend is 2 inches shorter than Noah. Can you tell if Noah’s friend is tall enough to go on the ride? Explain or show your reasoning.
List a possible height for Noah that would mean:
That his friend is tall enough to go on the ride.
That his friend is not tall enough for the ride.
On the number line, show all the possible heights that Noah’s friend could be.
Noah's friend is inches tall. Use and any of the symbols , , to express this height.
Activity Synthesis
Direct students’ attention to the reference created using Collect and Display. Ask students to share how they used the number line to show the possible heights and how they expressed the possible heights using symbols. 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 “height is greater than 60 or height is equal to 60” already on it and be updated with the more concise phrase “the height is greater than or equal to 60.”)
The purpose of this discussion is to introduce the symbols and . Ask selected students to share their response to the last question. They are likely to write something like “ or .” The and symbols are not enough to capture what we need here with a single mathematical statement. Introduce the new symbols and that mean “less than or equal to” and “greater than or equal to.” Invite students to use one of these new symbols to revise their answer to the last question. ( or ).
Lesson Synthesis
The purpose of this discussion is to compare how the and symbols are the same and different from the and symbols. Begin by displaying the inequalities and for all to see. For each inequality, discuss the following questions:
“What do we call this symbol?” (greater than or equal to; less than or equal to)
“How do we read this inequality statement?” ( is greater than or equal to 7. is less than or equal to 3.)
“How is this different from or ?” (With these inequalities, the value that is listed is not included as an acceptable value. For example, is not true, while is true.)
“On the number line, how would the graphs of and look the same or different?” (Both would have a circle at 7 and have an arrow pointing to the right. The graph of would have an open, or unfilled circle, to show that 7 is not a solution, while the graph of would have the circle filled in to show that 7 is a solution.)
Student Lesson Summary
Inequalities can be used to describe a range of numbers. Let’s say a movie ticket costs less than \$15. If represents the cost of a movie ticket, we can use to express what we know about the cost of a ticket.
Any value of that makes the inequality true is called a solution to the inequality.
For example, 5 is a solution to the inequality because (or “5 is less than 15”) is a true statement, but 17 is not a solution because (“17 is less than 15”) is not a true statement.
The inequality can be represented on a number line. The open circle at 15 shows that 15 is not a solution. The shading and arrow pointing left from 15 shows that all numbers less than 15 are solutions.
Here’s another example. In many places, people are eligible to get a driver’s license when they are at least 16 years old. If is the age of a person, then we can check if they are eligible to get a driver’s license by checking if their age makes the inequality (they are older than 16) or the equation (they are 16) true. The symbol , pronounced “greater than or equal to,” combines these two cases and we can just check if (their age is greater than or equal to 16).
The inequality can be represented on a number line. The closed, or filled in, circle at 16 shows that 16 is a solution. The shading and arrow pointing right from 16 shows that all numbers greater than 16 are also solutions.
Standards Alignment
Building On
Addressing
6.EE.A.2.b
Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity. For example, describe the expression as a product of two factors; view as both a single entity and a sum of two terms.
If students are having trouble interpreting the first three questions or articulating their responses, encourage them to make use of the number line that appears in the fourth question.
