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You can put some number of blocks on the left side and a different number on the right side and display to your students.

You might ask them what the image represents. Then have them share with a partner.

Then you might ask them to write an inequality statement to represent the image.

Finally, you might ask students to identify three different things they could do to make the scale balance.

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Transform static lesson delivery with game-based activities, formative assessments, and student collaboration.

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This provides 2 inequality apprentice tasks.

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Lesson 9 | Equations and Inequalities | 6th Grade Mathematics | Free Lesson Plan

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Write and graph inequalities for real-world conditions. (Part 1)

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https://www.openmiddle.com/solving-one-step-equations-greatest-solution/

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Directions: Use the digits 1 to 9, at most one time each, to create an equation where x has the greatest possible value.

Hint

Hint

How does the value of the number being added to x affect the value of x?
How does the value of the sum affect the value of x?
How can we tell whether it is better for the two missing numbers to have a larger or smaller value?

Answer

Answer

The equation 12 + x = 98 results in x having a value of 86.

Source: Robert Kaplinsky

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Solving One-Step Equations (Greatest Solution) - Open Middle®

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Directions: Use the digits 1 to 9, at most one time each, to create an equation where x has the greatest possible value. How does the value of the number being added to x affect the value of x? How does the value of the sum affect the value of x?

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https://tasks.illustrativemathematics.org/content-standards/6/EE/B/5/tasks/2203

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Providing instructional and assessment tasks, lesson plans, and other resources for teachers, assessment writers, and curriculum developers since 2011.

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https://www.geogebra.org/m/wygbm6wm

This has several activities that could be used for a hook, apprentice and expert task.

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Lesson Overview

This lesson is the first of a sequence of eight lessons where students learn to work with equations that have variables on each side. In this lesson, students recall a representation that they have seen in prior grades: the balanced hanger. The hanger is balanced because the total weight on each side, hanging at the same distance from the center, is equal in measure to the total weight on the other side.In the warm-up, students encounter two real hangers, one balanced and one slanted, and notice and wonder about what could cause the hangers' appearance. This leads into the first activity where students consider two questions about a balanced hanger: first, whether a change of the number of weights keeps the hanger in balance, and second, how to find the unknown weight of one of the shapes if the weight of the other shape is known. Students learn that adding or removing the same weight from each side is analogous to writing an equation to represent the hanger and adding or subtracting the same amount from each side of the equation. They reason similarly about how halving the weight on each side of the hanger is analogous to multiplying by 12 or dividing by 2. In both the hanger and the equation, these kinds of moves will produce new balanced hangers and equations that ultimately reveal the value of the unknown quantity.In the second activity, students encounter a hanger with an unknown weight that cannot be determined. This situation parallels the situation of an equation where the variable can take on any value and the equation will always be true, which is a topic explored in more depth in later lessons.As students use concrete quantities to develop their power of abstract reasoning about equations, they engage in MP2.

Ask your self

Ask students to share their ideas. Record and display the responses for all to see. In the interest of time, you can ask if anything students wondered was a "why" question, meaning the question begins with the word why. Refer to MLR 2 (Collect and Display).If not brought up during the first part of the discussion, ask students why they think the left hanger is unbalanced while the right hanger is balanced. Students should understand that a hanger will only balance if the weight of the unknown objects in both socks is the same. If they are not the same, then the heavier side is lower than the lighter side.

Lauching the Assignment

The purpose of this task is for students to understand and explain why they can add or subtract expressions from each side of an equation and still maintain the equality, even if the value of those expressions are not known. Both problems have shapes with unknown weight on each side to promote students thinking about unknown values in this way before the transition to equations.While the focus of this activity is on the relationship between both sides of the hanger and not equations, some students may start the second problem by writing and solving an equation to find the weight of a square. While students are working, identify those using equations and those not using equations to answer the second problem during the whole-class discussion. Launch Display the problem image for all to see. Tell students that this is a hanger problem similar to the one in the warm-up, only instead of the weights hidden inside socks, each block type represents a different weight. Give 5 minutes of quiet work time followed by a whole-class discussion. If using the digital activity, introduce the hanger problem to set the context and connection to the warm-up. Give students individual work time to figure out the weights and use the applet to check their work.

Question 2

Building on the previous activity, students now solve two more hanger problems and write equations to represent each hanger. In the first problem, the solution is not an integer, which will challenge any student who has been using guess-and-check in the previous activities to look for a more efficient method. In the second problem, the solution is any weight, which is a preview of future lessons when students purposefully study equations with one solution, no solution, and infinite solutions. The goal of this activity is for students to transition their reasoning about solving hangers by maintaining the equality of each side to solving equations using the same logic. In future lessons, students will continue to develop this skill as equations grow more complex culminating in solving systems of equations at the end of this unit.As students work, identify those using strategies to find the weight of one square/pentagon that do not involve an equation. For example, some students may cross out pairs of shapes that are on each side (such as one circle and one square from each side of hanger A) to reason about a simpler problem while others may replace triangles with 3s and circles with 6s first before focusing on the value of 1 square. This type of reasoning should be encouraged and built upon using the language of equations. Launch Arrange students in groups of 2. Give 5 minutes of quiet work time followed by partner discussion. Let students know that they should be prepared to share during the whole-class discussion, so they should make sure their partner understands and agrees with their solution.If students use the digital activity, the applet provides a way for students to check solutions. Encourage students to work individually (most likely they will need paper/pencil to work these problems) and then check their thinking using the digital applet. After students have had 5 minutes to work alone and with the applet, give them time to discuss their thinking with a partner before the whole-class discussion.

Summary

Select previously identified students to share their strategies for finding the unknown weight without using an equation. Ask students to be clear how they are changing each side of the hanger equally as they share their solutions.Next, record the equations written by students for each hanger and display for all to see in two lists. Assign half the class to the list for Hanger A and the other half to the list for Hangar B. Give students 1-2 minutes to examine the equations for their assigned hanger and be prepared to explain how different pairs of equations are related. The goal here is for student to use the language they developed with the hangers (e.g., "remove 6 from each side") on equations.For example, for Hanger A, you might contrast 3+6+6+6=4x+6 with 21+x=6+5x. Possible student responses:

Removing an x from each side of the second equation would result in the first equation.

x=3.75 grams makes both equations true.

You can subtract 6s from the sides of each equation and they are still both true.

For Hanger B, examining equations should illuminate why it is impossible to know the weight of the unknown shape. If we start with 6+6+x+x=x+x+3+3+6 and keep removing things of equal weight from each side, we might end up with an equation like 2x=2x. Any value of xwill work to make this equation true. For example if x is 10, then the equation is 20=20. It is also possible to keep removing things of equal weight from each side and end up with an equation like 6=6, which is always true.

"Summary 2"

The purpose of this discussion is to have students revisit the warm-up and connect it to the activities, reflecting on why the hanger is an appropriate and helpful analogy for an equation.Ask these questions:

"In the warm-up we wondered why one hanger was slanted, whether there were weights in one blue sock that made it heavier than the other, whether the crooked hanger would straighten out if another sock was added to the other side (add any other pertinent things your students wondered). How would you answer these questions now?"

"What is an equation? What does the equal sign in an equation tell you?" (An equation is a statement that two expressions have the same value. The equal sign tells you that the expressions on either side must have the same value, however that value is measured-as a count of objects, a measurement like 10 miles or 6 seconds, or numbers without units.)

"What features do balanced hangers and equations have in common?" (Both representations have sides that are equal in value, even if the actual value of a side is unknown. Each side can contain numbers we do not know in the form of either shapes or variables. Changing the value of one side of a hanger or equations means changing the value of the other side by the same amount.)

"You saw an example of a hanger where the unknown weight could not be determined. Can you design your own hanger like this one? How would you think about the weights needed on each side?" (If students completed the extension, you might ask them to also design a hanger with no solution.)

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We would use slide 11 as the expert tasks, allowing students to work through the first ten slides as a review, but then NOT moving on beyond 11 at this point in the unit.

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In this activity, students explore linear inequalities and make connections among multiple representations (including algebraic expressions, verbal statements, number line graphs, and solution sets). RELATED ACTIVITIES This is the first activity in a series of three. Here are links to the others: 2. Compound Inequalities on the Number Line https://teacher.desmos.com/activitybuilder/custom/57b7f808ce9904f705558c4e 3.

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Andrew Stadel asked: What amount per week did Woody settle for?? What is the first question that comes to your mind?

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inequality milestone

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City Cleanup [Day 1] | Algebra: Inequalities 3 Act Math Task

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Algebra lesson investigating inequalities by solving, verifying, and graphing solutions.

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Found on CFWV.com and there are literally thousands of similar career-related math activities waiting to be found!

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Real-Life Math

You are a forensic psychologist with a private practice in a large city. Your expertise lies in understanding the human mind as it relates to criminal matters.

Up until one month ago, you worked for a psychiatric hospital. You treated criminals who had been declared mentally unfit. You tested them when they were first admitted, provided treatment and then regularly re-tested them to see if they were ready to leave the hospital.

In addition, you were frequently called upon to testify at parole hearings. After examining a prisoner, you would state whether they should be released from prison on parole.

Six months ago, you decided to quit working at the hospital and go into private practice. You did so because you wanted to work on a greater variety of cases. Today it is officially a month since you launched your solo practice.

Judging from the number of assignments you've had during the first month, you made the right decision. Now, a number of individuals and organizations seek out your expertise, including lawyers, prosecutors, courts and prison administrators.

While you enjoy the greater variety of cases, there are some features of being self-employed that you could do without. Now that you're in private practice, you're also in charge of all the paperwork that comes with running a business, such as billing clients. Sometimes it can be a little overwhelming.

It's Friday evening -- time to write up the monthly invoices. It's been a busy 4 weeks. In addition to administering 12 intelligence tests, you spent 20 hours helping a prosecutor prepare for trial, and another 10 hours in court testifying in 2 child custody cases. In addition, you examined a prisoner who will be coming up for parole soon.

Before figuring out what each client owes, you want to figure out what your total earnings were for the month. Normally, you charge $80 per hour. However, you charge $50 per hour when testifying in court, and a flat rate of $350 for an IQ test.

What are your total earnings for the month?

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College advice, college admission applications, pay for college, financial aid, scholarships, apply online for admission, online applications at Xap.

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https://cfwv.com/Career_Planning/Career_Profile/Career_Profile.aspx?id=eyu0dops5EUfXAP2BPAXJRwmDtrKgXAP3DPAXXAP3DPAX&screen=10&ActivityType=M

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Real-Life Math

You are a forensic psychologist with a private practice in a large city. Your expertise lies in understanding the human mind as it relates to criminal matters.

Up until one month ago, you worked for a psychiatric hospital. You treated criminals who had been declared mentally unfit. You tested them when they were first admitted, provided treatment and then regularly re-tested them to see if they were ready to leave the hospital.

In addition, you were frequently called upon to testify at parole hearings. After examining a prisoner, you would state whether they should be released from prison on parole.

Six months ago, you decided to quit working at the hospital and go into private practice. You did so because you wanted to work on a greater variety of cases. Today it is officially a month since you launched your solo practice.

Judging from the number of assignments you've had during the first month, you made the right decision. Now, a number of individuals and organizations seek out your expertise, including lawyers, prosecutors, courts and prison administrators.

While you enjoy the greater variety of cases, there are some features of being self-employed that you could do without. Now that you're in private practice, you're also in charge of all the paperwork that comes with running a business, such as billing clients. Sometimes it can be a little overwhelming.

It's Friday evening -- time to write up the monthly invoices. It's been a busy 4 weeks. In addition to administering 12 intelligence tests, you spent 20 hours helping a prosecutor prepare for trial, and another 10 hours in court testifying in 2 child custody cases. In addition, you examined a prisoner who will be coming up for parole soon.

Before figuring out what each client owes, you want to figure out what your total earnings were for the month. Normally, you charge $80 per hour. However, you charge $50 per hour when testifying in court, and a flat rate of $350 for an IQ test.

What are your total earnings for the month?

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College advice, college admission applications, pay for college, financial aid, scholarships, apply online for admission, online applications at Xap.

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Desmos activity Weight for It

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Students use reasoning, equations, and tape diagrams to determine unknown weights on a see-saw.

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Performance Task: Planning a School Fair Fundraiser

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Performance Task: Planning a School Fair Fundraiser

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Performance task generated through ChatGPT

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https://www.map.mathshell.org/lessons.php?unit=6210&collection=8

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Mathematical goals

This lesson unit is intended to help you assess how well students are able to use an approximate mathematical model of a real-world situation and use data to test, critique, and compare models. In particular this unit aims to identify and help students who have difficulty using variables to represent quantities and analyze the relationship between these variables using tables, graphs, and equations.

Introduction

The lesson unit is structured in the following way:

Before the lesson, students attempt the Car Skid Marks task individually. You then assess their responses and formulate questions that will prompt them to review their work.

At the start of the lesson, students think individually about their responses to the questions set. They then work in small groups to combine their thinking to produce a collaborative solution to the Car Skid Marks task, in the form of a poster.

Working in the same small groups, students evaluate and comment on sample responses, identifying strengths and weaknesses in each approach and comparing them with their own work. In a whole-class discussion, students compare and evaluate the methods they have seen and used.

In a follow-up lesson, students reflect on their work and what they have learned.

Materials required

Each student will need a copy of the Car Skid Marks task, the How Did You Work? questionnaire, a mini-whiteboard and eraser, and some plain or squared paper to work on. Calculators and graph paper should be available on request.

Each small group of students will need a large sheet of paper and enlarged copies of the Sample Responses to Discuss.

Time needed

20 minutes before the lesson, a 100-minute lesson (or two 55-minute lessons), and 15 minutes in a follow-up lesson (or for homework). Exact timings will depend on the needs of the class.

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This lesson unit is intended to help you assess how well students are able to use an approximate mathematical model of a real-world situation and use data to test, critique, and compare models. In particular this unit aims to identify and help students who have difficulty using variables to represent quantities and analyze the relationship between these variables using tables, graphs, and equations.

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In the last row the students can create their own word problem to simplify.

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https://www.map.mathshell.org/download.php?fileid=1604

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Home> Grade 6> Expressions & Equations> Solving One-Step Inequalities with Addition

Directions: Using the digits 0 to 9 at most one time each, place a digit in each box to create an inequality with a solution of x<4

Hint

Begin by trying two of the digits to see what happens.

Answer

The possible answers are:
x+0<4
x+1<5
x+2<6
x+3<7
x+4<8
x+5<9

Source: Annie DeAngelo

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Solving One-Step Inequalities with Addition | Open Middle®

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Home> Grade 6> Expressions & Equations> Solving One-Step Inequalities with Addition Directions: Using the digits 0 to 9 at most one time each, place a digit in each box to create an inequality with a solution of x<4 Begin by trying two of the digits to see what happens.

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https://teacher.desmos.com/activitybuilder/custom/6151e78476f65cfb1c349218?r=w.hd&collections=6205963378162a2045079529%2C5e72b4b9feeb100f56bcac4f

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Students use tables, graphs, and equations to help customers make decisions about what type of transportation ticket to buy.

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In this task, 8.1 would be your hook. The next two parts (8.2 and 8.3) are truly apprentice tasks.

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An inequality tells us that one value is less than or greater than another value.

Suppose we knew the temperature is less than \(3^\circ \text{F}\), but we don't know exactly what it is. To represent what we know about the temperature \(t\) in \(^\circ \text{F}\) we can write the inequality: \(\displaystyle t< 3\)

The temperature can also be graphed on a number line. Any point to the left of 3 is a possible value for \(t\). The open circle at 3 means that \(t\) cannot be equal to 3, because the temperature is less than 3.

Here is another example. Suppose a young traveler has to be at least 16 years old to fly on an airplane without an accompanying adult.

If \(a\) represents the age of the traveler, any number greater than 16 is a possible value for \(a\), and 16 itself is also a possible value of \(a\). We can show this on a number line by drawing a closed circle at 16 to show that it meets the requirement (a 16-year-old person can travel alone). From there, we draw a line that points to the right.

We can also write an inequality and equation to show possible values for \(a\): \(\displaystyle a>16\) \(\displaystyle a=16\)

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Illustrative Mathematics - Students | Kendall Hunt

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An inequality tells us that one value is less than or greater than another value. Suppose we knew the temperature is less than \(3^\circ \text{F}\), but we don't know exactly what it is.

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Again, the first activity is the hook. The second and third activities in this task are Apprentice tasks.

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Let's say a movie ticket costs less than \$10. If \(c\) represents the cost of a movie ticket, we can use \(c < 10\) to express what we know about the cost of a ticket.

Any value of \(c\) that makes the inequality true is called a solution to the inequality.

For example, 5 is a solution to the inequality \(c < 10\) because \(5<10\) (or "5 is less than 10") is a true statement, but 12 is not a solution because \(12<10\) ("12 is less than 10") is not a true statement.

If a situation involves more than one boundary or limit, we will need more than one inequality to express it.

For example, if we knew that it rained for more than 10 minutes but less than 30 minutes, we can describe the number of minutes that it rained (\(r\)) with the following inequalities and number lines. \(\displaystyle r > 10\)

\(\displaystyle r < 30\)

Any number of minutes greater than 10 is a solution to \(r>10\), and any number less than 30 is a solution to \(r<30\). But to meet the condition of "more than 10 but less than 30," the solutions are limited to the numbers between 10 and 30 minutes, not including 10 and 30.

We can show the solutions visually by graphing the two inequalities on one number line.

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Illustrative Mathematics - Students | Kendall Hunt

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Let's say a movie ticket costs less than \$10. If \(c\) represents the cost of a movie ticket, we can use \(c 10\) \(\displaystyle r 10\), and any number less than 30 is a solution to \(r<30\).

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10.1 Hook Task 10.2 & 3 Apprentice Tasks

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When we find the solutions to an inequality, we should think about its context carefully. A number may be a solution to an inequality outside of a context, but may not make sense when considered in context.

Suppose a basketball player scored more than 11 points in a game, and we represent the number of points she scored, \(s\), with the inequality \(s >11\). By looking only at \(s >11\), we can say that numbers such as 12, \(14\frac12\), and 130.25 are all solutions to the inequality because they each make the inequality true.
In a basketball game, however, it is only possible to score a whole number of points, so fractional and decimal scores are not possible. It is also highly unlikely that one person would score more than 130 points in a single game.

In other words, the context of an inequality may limit its solutions.

Here is another example:

The solutions to \(r<30\) can include numbers such as \(27\frac34\), 18.5, 0, and -7. But if \(r\) represents the number of minutes of rain yesterday (and it did rain), then our solutions are limited to positive numbers. Zero or negative number of minutes would not make sense in this context.

To show the upper and lower boundaries, we can write two inequalities:

Inequalities can also represent comparison of two unknown numbers.

Let's say we knew that a puppy weighs more than a kitten, but we did not know the weight of either animal. We can represent the weight of the puppy, in pounds, with \(p\) and the weight of the kitten, in pounds, with \(k\), and write this inequality: \(\displaystyle p >k\)

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Illustrative Mathematics - Students | Kendall Hunt

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When we find the solutions to an inequality, we should think about its context carefully. A number may be a solution to an inequality outside of a context, but may not make sense when considered in context.