1. Introduction to Inequalities in the Real World

2. Lesson Overview (1 of 4) Lesson Objective
SWBAT translate real-world situations into mathematical statements using inequalities and variables.
Language Objective: Students will define key terms like, variable, inequality and infinitely many and give examples of real-world situations that have infinitely many answers.
Lesson Description
After completing a warm-up that reviews 6EEB5 ( , we will engage students in learning by watching a rollercoaster video and using this as a way to discuss inequalities. In a pair-share activity, students will explore riders’ heights for this rollercoaster. Students will discover that we can have multiple (and ultimately infinite) solutions to these real-world problems. This example will launch students into writing inequalities in real-world situations and introduce key words like inequality, variable and infinity. Finally, students will practice as a whole class writing inequalities, and will then work in pairs on a class work assignment before explaining Buzz Lightyear’s signature phrase, “To infinity and beyond” in a writing exercise.

3. Lesson Overview (2 of 4) Lesson Vocabulary
A variable is a symbol that Is used to represent an unknown number. The letters x or y are examples of variables.
An inequality is a symbol, like > or <, that states that two values are NOT equal.
Materials
Scaffolded Class notes
Roller Coaster Video:
Common Core
State Standard
CCSS.Math.Content.6.EE.B.8 Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.

4. Lesson Overview (3 of 4) Scaffolding
This lesson focuses on inequality vocabulary and can be scaffolded for ELL and SPED students. Teacher can spend time working on flashcards or foldables which focus on synonyms and symbols for greater than and less than. The scaffolded notes provided help learners to follow along with the lesson, and the scaffolded notes with answers can be provided to students with this accommodation.
Enrichment
One way to extend this lesson is to include more complicated inequality vocabulary terms. For example, teachers can change the Launch question to include words like “at least” or “at most” rather then “less then” or “more then.”
Online Resources for Absent Students
These are a series of 3 videos covering inequalities and writing inequalities in real-world situations:

5. Lesson Overview (4 of 4) Before and After
This is a critical lesson in the larger context of algebra and mathematics because we are introducing the idea of infinity and multiple answers. Prior to this lesson, there is usually one correct answer to a real-world situation. However, students will learn in this lesson that there can be many (and actually infinitely many) solutions to real-world problems. This shift in thinking represents a critical change in mathematical thinking.
Topic Background
This lesson represents an important theoretical change in how students think about answering math questions. In this lesson, we introduce the idea of infinity and multiple answers. The deeper-level thinking behind these problems asks students to reason abstractly in a way that they have not done before. This sophisticated way of representing mathematics links with mathematical practices #2:

6. Warm Up 2x + 6 > 12 2(6) + 6 > 12 12+ 6 > 12 18 > 12
OBJECTIVE: SWBAT translate real-world situations into mathematical statements using inequalities and variables. Language Objective: Students will define and give examples of real-world situations that have infinitely many answers.
A set has the following values: {2, 4, 6}. Which of the values in this set make the inequality below true? Show your work!
2x + 6 > 12
Hint: Use substitution!
2x + 6 > 12
2(6) + 6 > 12
12+ 6 > 12
18 > 12
2x + 6 > 12
2(2) + 6 > 12
4 + 6 > 12
10 > 12
2x + 6 > 12
2(4) + 6 > 12
8 + 6 > 12
16 > 12
(5 min) Time passed 5 min
In-Class Notes
This warm-up links directly with CCSS 6EEB5 where students are required to “Use substitution to determine whether a given number in a specified set makes an equation or inequality true.”
You can provide students with the pop up hint to remind them to use substitution.
If students knowledge of inequalities is limited, you can also change the problem into an equation and make one of the values in the given set a 3.
False!
True!
True!
The values 4 and 6 make the inequality true!
Agenda

7. Agenda: 1) Warm Up (Independent)
OBJECTIVE: SWBAT translate real-world situations into mathematical statements using inequalities and variables. Language Objective: Students will define key terms like, variable, inequality and infinitely many and give examples of real-world situations that have infinitely many answers.
1) Warm Up (Independent)
2) Launch – Ride Time and Pair-Share (Whole Class and Pairs)
3) Explore - Writing Inequalities (Whole Class)
4) Summary – Vocabulary Overview (Pairs)
5) Practice – Class work (Pairs)
6) Assessment – To Infinity and Beyond! (Independent)
(1 min) Time passed 6 min
In-Class Notes
You can review the agenda with students and go over the objective before starting the launch.
Students can read the agenda and objective, too, while you prep the video for the next slide.

8. Let’s go for a ride: Are you ready?!?! Launch Agenda
(3 min) Time passed 9 min
In-Class Notes
There is one video of someone on a roller coaster provided when you click on “Are you ready?!?” – this is through youtube:
Any video of someone on a roller coaster will work – this is the hook to get students motivated for the mathematics of being a certain height to ride a roller coaster.
Preparation Notes
Preview the video and make sure you have access to youtube.com – any video of roller coaster will work
There is a 5 second advertisement which if you load first, you can skip and then immediately show the video.
Are you ready?!?!
Agenda

9. Launch Continued: Pair-Share
Mei and Erika are at Six Flags with their families. They see the following sign in front the Dropping Dragon Roller Coaster. Mei is 56 inches tall and Erika is 49 inches tall.
Your height must be greater than 50 inches to ride the Dropping Dragon!
(5 min) Time passed 14 min
In-Class Notes
After watching the roller coaster video, students can work in pairs on this pair-share activity about Mei and Erika. The first two questions are usually quickly completed by students, and the third question allows for student discussions and comparison of answers.
You can ask students to write on the board possible heights for Miguel. This will help to illustrate to students that we can have many possible heights.
One way to extend this would be to ask students to not just write a height as a whole number (by inches) – this will help to lead of a discussion on infinitely many (i.e. Miguel could be 42.3 inches or 42.4 inches, etc.)
Another possible extension is to ask students to give their height not in inches but in feet – converting units may be a topic that you have covered in class or not
Agenda

10. Launch Continued:
Who will be able to go on the ride? Why?
Who will NOT be able to go on the ride? Why not?
Miguel also wants to go on the Dropping Dragon. What is one height that he could be in order to ride the roller coaster?
Mei will be able to ride the Dropping Dragon
because she is taller than 50 inches.
Erika will be not able to ride the Dropping
Dragon because she is shorter than 50 inches.
Your height must be greater than 50 inches to ride the Dropping Dragon!
(2 min) Time passed 16 min
In-Class Notes
Students can lead the conversation to review answers, and in particular, should take the lead to answer Miguel’s question.
You can start to emphasize that there are a LOT of heights that Miguel could be – you may even want to prompt students, how many different heights can we list for Miguel?
Make sure to mention decimal/fraction heights to students sometime in this slide – or ask students a leading question, is everyone exactly 55 inches tall or 56 inches tall?
Agenda

11. Launch Continued: Pair-Share
Your height must be less than 50 inches tall to ride the Mini-Coaster!
Mei’s little brother, Sai, wants to go on the Mini-Coaster ride. Sai is 34 inches tall. Mei sees the following sign in front of the ride.
Can Sai go on the Mini-Coaster? Why or why not?
Can Mei go on the ride with him? Why or why not?
Erika’s little brother, Nick, wants to go on the Mini-Coaster. What is one height that he could be to go on this ride?
Mei is 56 inches tall.
(2 min) Time passed 18 min
In-Class Notes
This is a similar set of questions as the prior slides, and will emphasis the comparison/contrasting nature of less than or greater than.
Again students can work in pairs to brainstorm Nick’s possible heights.
You can extend this activity by asking students to list more than 1 height for Nick, or ask them to give heights in decimal form.
Agenda

12. Launch Continued: Pair-Share
Your height must be less than 50 inches tall to ride the Mini-Coaster!
Mei’s little brother, Sai, wants to go on the Mini-Coaster ride. Sai is 34 inches tall. Mei sees the following sign in front of the ride. Can Sai go on the Mini-Coaster? Why or why not?
Can Mei go on the ride with him? Why or why not?
Erika’s little brother, Nick, wants to go on the Mini-Coaster. What is one height that he could be to go on this ride?
Sai can go on the Mini-Coaster because he
is shorter than 50 inches.
Mei can NOT go on the Mini-Coaster because she
is taller than 50 inches.
(2 min) Time passed 20 min
In-Class Notes
Again, students can lead the review of these questions, and will hopefully be able to come up with a host of heights for Nick.
If possible, you may want to leave a list of all the heights for Nick on your slideshow – we can use this to help illustrate the idea of infinitely many which will be defined in slide #22.
This links with math practice #6 which encourages students to examine and critique others work.
Agenda

13. Launch Continued: Turn and Talk
Compare the two signs for the Dropping Dragon and the Mini-Coaster.
Explain 1 similarity to your partner
Explain 1 difference to your partner
Your height must be greater than 50 inches to ride the Dropping Dragon!
Your height must be less than 50 inches tall to ride the Mini-Coaster!
(2 min) Time passed 22 min
In-Class Notes
This slide can be discussed as a whole class after students have a chance to turn and talk.
The key point of this slide is to highlight the difference between greater than and less than, and lead students into the inequality symbols for these.
Agenda

14. Explore: Let’s Review:
How can we represent this sign as a mathematic statement?
Your height must be greater than 50 inches to ride the Dropping Dragon!
Let’s Review:
What is a variable?
A variable is a symbol that Is used to represent an unknown number.
An example of a variable is the letter x or y.
In the sign above, what number is unknown?
(1 min) Time passed 23 min
In-Class Notes
Depending on prior lessons that you have taught, this definition may take 1 minute to review with students, and we can ask students what is the most popular variable!
If this is prior knowledge for students, you can illicit from students the definition for variable. And ask students what is the unknown number.
The person’s height!
We can use x to represent the unknown height of any rider!
Agenda

15. to ride the Dropping Dragon!
Explore:
How can we represent this sign as a math statement? x
>
Your must be
to ride the Dropping Dragon!
height
greater than
50 inches
What symbol connects x with 50 inches?
An equal sign?
(2 min) Time passed 25 min
In-Class Notes
You can use the animation to highlight how each part of the sign turns into part of the math statement.
The dragon helps to emphasis that we know that we can’t use an equal sign since we need riders with heights of more than 50 inches.
The use of inequalities in math statements really represents an important change in student thinking – and you can tell students this! We are now getting into high school math, we don’t just have = signs – we have lots of other symbols to represent more complex mathematical situations.
Greater than can be represented using a symbol!
Agenda

16. Explore: What is an inequality?
Greater than
Less than
>
<
The greater than and less than symbols are called inequalities!
An inequality is a symbol, like > or <, that states that two values are NOT equal.
What are the symbols for greater than and less than?
How can we remember which one is less than?
Take your left hand, hold it up and make and “L” , like the picture
(2 min) Time passed 27 min
In-Class Notes
This slide may be review for some students but provides an opportunity to define inequality again (and compare this with an equal sign) and also give students a sneaky way to remember less than using their hands.
Students can fill in the definition of inequality using their class notes.
You can also ask all students to demonstrate the left hand trick with their hands, and ask a follow up – how can you remember your greater than sign?
Now, close your hand a little
Do you see a less than sign?
Agenda

17. to ride the Dropping Dragon!
Explore:
Let’s review!
How can we represent this sign as a mathematical statement? x
>
50 inches
Your must be
to ride the Dropping Dragon!
height
greater than
Step 1: We find the unknown value.
50 inches
Step 2: We pick a variable.
Step 3: We find the number connected to
the variable.
Step 4: We use an inequality to
connect the variable and number.
(3 min) Time passed 30 min
In-Class Notes
Students can write in the missing parts to their steps on student notes, and also follow along with how we have written the sign as a math statement.
Students will have a chance to use these steps themselves in the next slide.
Agenda

18. tall to ride the Mini-Coaster!
Explore: You try!
How can we represent this sign as a mathematical statement? x
<
Your must be
tall to ride the Mini-Coaster!
height
less than
50 inches
Review Question:
What heights (or values for x) will make that inequality statement above true?
(6 min) Time passed 36 min
In-Class Notes
Students can try independently or with a partner how to write this sign as a math statement.
You can click in the review questions after all students have the correct inequality written.
The review questions set-up the next key term for this lesson which is infinitely many – so we may want to even take time to brainstorm as many heights as we can that make the inequality true – or you can click back to the slide with Erika’s brother Nick
Is there just 1 answer to this problem?
Agenda

19. For this inequality statement, we have many, many solutions!
Explore:
x > 50
For this inequality statement,
we have many, many solutions!
In fact, we have infinitely many solutions for an inequality statement like x > 50.
infinitely many solutions, means that we have never-ending answers that will make a math statement true.
(2 min) Time passed 38 min
In-Class Notes
We can use the last question in the previous slide to introduce this more formal definition. Students will have this definition already written in class notes and will just need to fill in “never-ending”
Agenda

20. Summary:
A variable is a symbol that Is used to represent an ____________ number. A variable is represented by a letter like _____ or ______.
unknown x y
An inequality is a symbol, like ____ or ____, that states that two values are NOT equal.
>
<
infinitely many solutions , means that we have never-ending answers that will make a math statement _____.
true
(2 min) Time passed 40 min
In-Class Notes
To review all of the important vocab for this unit, students can fill in the answers by raising their hands.
Note that the variable key word may have answers besides x and y.
Agenda

21. Practice:
Write an inequality statement using the variable x to represent each real-world situation below. Then, write 3 possible solutions to each inequality.
Water freezes at any temperature less than 0 degrees Celsius (°C).
x < 0 °C
3 possible solutions: -5 °C, -20 °C, -52 °C
(2) Kiera’s weekly allowance is greater than $10.
x > $10
(5 min) Time passed 45 min
In-Class Notes
In this slide, you can guide students through the first example about water freezing, and also solicit from students possible degree measurements. There have been 3 included in the slide but can be omitted, if desired.
Students can try the second example about Kiera’s allowance on their own or work with a partner. Again, we want to reinforce the idea of infinitely many solutions and that we are only providing 3 examples.
The last example is a great opportunity to have students critique each other’s work – Math Practice #3 – by asking students to present their work at the board and provide feedback on whether they agree or not with their peers.
(3) In his job as a lawyer, Cameron works more than 50 hours
per week.
x > 50 hours
Agenda

22. Practice - Classwork Agenda (7 min) Time passed 52 min In-Class Notes
Students can work in pairs or small groups on this class work assignment and when done write their answers on the board.
To cover the CCSS math practices, if you have time, it would be great to have students explain and critique each others’ work before showing the answers on the next slide.
If teachers do not have paper copies for students of these class notes, they may want to project onto an overhead or copy each slide separately onto the overhead since the font is small. However, hand-outs for this assignment are strongly recommended!
Preparation Notes
Class work assignment for students
Agenda

23. Practice – Classwork Answers
(3 min) Time passed 55 min
In-Class Notes
You can review the class work problems with students. Question #6 about Malik is an important to review with students because it covers the specific case of 5 > 5 or inclusion when we have just greater than and just less than (with no equal to).
If you can leave time to review #6 that’
Preparation Notes
Class work assignment for students
Agenda

24. Assessment In the Toy Story movies, Buzz Lightyear has a saying,
“To Infinity and Beyond.”
Infinity is a similar word to infinitely many. Write a sentence explaining how these words are related.
(5 min) Time passed 60 min
In-Class Notes
I would suggest that you collect this exit ticket as a writing assignment or ask students to share what they’ve written with the class (if time remains).
This exit ticket will allow teachers to formatively assess whether their students understand the idea of infinity or not – since we have introduced this in the lesson, this exit ticket will hopefully connect students with other real-world experiences that they already know about infinity.
You may want to ask students to give other real-world examples of infinity or things that go on forever – this deepens student understanding and helps students to connect ideas with today’s lesson and the CCSS 6EE8.
Agenda

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