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https://shorts.flipgrid.com/watch/22669209200231016

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Get to know a little bit about Ms. DeLuca! by Deanna D

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Overall impressions: This platform is second to Teams. Zoom does not require you to be in a certain district, and it has almost all of the functionalities of Teams. Zoom may be designed more for businesses than schools.
Pros: Anyone can access Zoom, even without an account. After you record a session, it is available online at all times and downloadable. Zoom is very user friendly and participants are capable of sharing their screen or documents. There is also a handy whiteboard function.
Cons: Without a school account or premium account, sessions cannot last longer than 45 minutes. There are less regulations on Zoom.

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Zoom_Group 1 (DeLuca, Jackosn, Hagan, Reid).mp4

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Overall impression: It's clear that Blackboard Collab, like Microsoft Teams, was designed to be geared more towards students and education. It is a useful tool for math classes. It's main downfall is that it is not free unless your school district pays for it.
Pros: Blackboard Collab has many functions and is very user-friendly. Students and teachers can share their screen, documents, pdfs, etc., use the white board function, post questions in the chat, raise their hand using a click of a button, and easily record the meeting.
Cons: It is not free. We had access to all of these functions through the University of South Carolina, but Blackboard Collab is expensive and not widely used in school districts.

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BB Colaborate_Group 1 (DeLuca, Jackosn, Hagan, Reid).mp4

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Overall impression: Teams is my top choice for a virtual collaboration platform. Most districts are either Google-based or Microsoft-based, so if your school supports Microsoft, this is the way to go. It was clearly built with students in mind. This platform was also the most versatile. Along with the many functionalities, students and teachers are given a recorded, live transcript, can rearrange the classroom, and split off into breakout groups.
Pros: This platform was also the most versatile. Along with the many functionalities, students and teachers are given a recorded, live transcript, can rearrange the classroom, and split off into breakout groups.
Cons: Microsoft Teams may have too many functions that could distract students.

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Teams_Group 1 (DeLuca, Jackosn, Hagan, Reid).mp4

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Overall impressions: Google Meet is most convenient for quick meetings, but it is not idea for class or profession meetings.
Pros: Going to meet.google.com and starting a meeting is about as easy as it gets. Students have multiple ways to access a meeting and the platform is compatible with Google Classroom.
Cons: The only accessory besides screensharing that is available without premium is the whiteboard. You cannot record meetings or go into breakout rooms.

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Overall impressions: Teams, if the district supports it, is a great way to be collaborative with students. It lends itself well to group projects, storing notes and old recordings, and scheduling.
Pros: It has a ton of features. A student could spend their entire day in Microsoft Teams without having to leave. The classrooms are extremely customizable and teachers can add whatever they need for the class.
Cons: It has a ton of moving parts. It is easy to get lost navigating the never-ending features.

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Overall impressions: Like Google Meet, Classroom is a Google companion that is designed to be concise. It is a great tool for teachers to get information and assignments out to students, and it is very compatible with Google Drive.
Pros: Teachers can schedule posts for certain times. Since it's compatible with Google Forms, assignments are very easy to create. The grades can automatically be sent to the gradebook.
Cons: It's very simple. While it's compatible with a lot of external resources, Classroom itself does not have to many options for customization.

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Overall impressions: Discord is a fun, creative take on a class platform. I'm not sure it's ready for the public school system yet.
Pros: It has great chat and video features that are typically well-suited for video games and clubs.
Cons:  There does not seem to be a great way to fully monitor what students would say in these chats. There's no way to prevent them from chatting during class, for example, without using an additional monitoring software.

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Overall impressions: If Blackboard was cheaper and more available, this would be my second choice of class platform next to Microsoft Teams. It has sounds organization and tons of capabilities.
Pros: The video conferences platform, mentioned previously, is great. Students can see current and past classes, grades, and even sometimes assignments. The assessments/submissions are the highlight of the platform. They're extremely customizable. Also the discussion boards are a great way to utilize Blackboard.
Cons: Blackboard has its flaws. Sometimes pages do not load correctly and there's constant maintenance. Also, if the district does not support Blackboard, it can be very expensive to use.

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Overall impressions: ClassFlow is a newer class platform brought to us by the creators of the Promethean board. It still has kinks to work out, but it has a lot of unique interactive elements.
Pros: Students can scan a QR code and immediately access a live activity. For example, students can scan the code and then all write a word that comes to mind when they hear "triangle." Then, all of the student answers are uploaded and the class can connect words in a mapping activity.
Cons: It is newer than other platforms. Even in the demonstration in class, there were certain elements that did not work like they should have.

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Overall impressions: Khan Academy is more than awesome review videos and content activities. It can also act as a class platform.
Pros: You can assign videos and assignments to students in your class. As a teacher, you can easily monitor their progress. This platform is most useful for virtual learners since the teachers can craft which videos are included.
Cons: It is not a very diverse platform. It is limited in its capabilities, especially when compared to platforms like Blackboard and Microsoft Teams.

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Overall impressions: Polygraph is an awesome Guess Who-like activity that challenges students to correctly narrow down graphs to the correct one. It is extremely verstile and can be incorporated at any grade level.
Pros: There are plenty of pre-made activities for students. Teachers can also make their own Polygraph activity based on the grade level and lesson. These activities can either stand-alone or integrated into a Desmos activity.
Cons: I have no cons for Polygraph and would always recommend it.

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Overall impressions: These pre-set assessments are a great way to check for understanding and get student feedback instantly.
Pros: These pre-set assessments are 100% customizable and can be built into any lesson. The teacher can also revise the assessment and answer key accordingly.
Cons: Desmos will not automatically export grades to Google Classroom or Microsoft Teams like you can with other applications.

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Desmos Sample Assessment (Paper)
Desmos Midterm - Companion Assessment (Pape)

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Overall impressions: Computation Layer is user-friendly, easy to learn, and extremely impactful. I personally have used Computation Layer a bunch in my Desmos Activities, even if it's "only" for the initial text function.
Pros: Computation Layer has tons of pre-made starter slides, add-ons, and useful touches to make lessons unique. Teachers can either copy the parts of code they like or copy entire slides. Every code in Computation Layer has an example slide so teachers can see exactly what the code does.
Cons: Teachers can be overwhelmed if they are not familiar with Computation Layer or are apprehensive to code.

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Overall impressions: This was my "from scratch" lesson that I used at my placement. The students really enjoyed the cart sorting activities and the fun-fact starter slide. It was easy to make the lesson my own. I used the Teacher-view to monitor my students' work and facilitate discussions.

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5.7: Converting Forms of Linear Equations

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Objective:
Rewrite linear equations in standard, slope-intercept, and point-slope form.

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Overall impressions: Making a lesson on Geogebra is fairly straight forward. For most lessons, I much prefer Desmos. The Geogebra model is neat since you can embed other people's homemade materials, like the unit circle.
Pros: The pros include easy access for students, customizable layouts/sizes, and then thousands of pre-made activities.
Cons: Teachers cannot directly monitor students' work like they can on Desmos. The layout is more similar to a worksheet whereas Desmos is closer to an actual lesson.

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GeoGebra Classroom Demo (Start to Finish)

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Overall impressions: This is an awesome tool for all ages. It especially becomes beneficial for high school students in pre-calculus and calculus when graphs get more difficult to draw by hand
Pros: The axes and curves are color-coded and can be labeled. Students can also clearly see the shaded areas when dealing with integrals or inequalities. 3-D shapes also work well.
Cons: The functionalities can be overwhelming for new users. Students would definitely need a run through of how to use the applications.

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Free online 3D grapher from GeoGebra: graph 3D functions, plot surfaces, construct solids and much more!

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Overall impressions: I wish I had this in 531! It makes constructions so much easier. I think this would definitely help students connect with the material.

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Overall impressions: SharePoint is a member of the Microsoft family and so it is seemlessly compatible with Microsoft Teams. It is a good platform for sharing and posting.
Pros: SharePoint is a highly versitile platform, like Teams. It allows for creativity and originality. It's more than a cloud service like OneDrive or GoogleDrive.
Cons: It takes a bit of effort to use. Teachers may be apprehensive to use this over other platforms because of the learning curve. A lot of features and options make SharePoint confusing at first.

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Overall impressions: JamBoard is an awesome, easy to use tool that encourages collaboration in and out of the classroom. I would strongly recommend this be used with any group.
Pros: Teachers can set up their JamBoard slides to prepare ahead of time. Students then click on the link and are free to collaborate (draw, write, post, etc). Teachers can also use this for group work that gets presented at the end of class.
Cons: I have no cons. JamBoard is really a great tool that can be used at any level.

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Overall impressions: Annotate is a cool, useful tool. It is a Google Chrome extension that allows you to write on the screen and save your notes.
Pros: This is obviously useful for a math class. Teachers can export their annotations to Google Classroom, Teams, etc. so students also have a copy of the notes to refer to.
Cons: You do need to create an account in order to use this extension. There appear to be occasional glitches, but nothing that would prevent me from endorsing this tool.

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Annotate | Mobile Interactive Whiteboard, Student Response System

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Annotate integrates a Mobile Interactive Whiteboard, Screen Mirroring, Formative Assessments, and Collaboration on your tablet.

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Overall impressions: Screencastify is easy to use, compatible with platforms like Google Classroom, and is extremely versatile. It was clearly made for the classroom.
Pros: Teachers can create assignments on Screencastify and link the assignment up to Google Classroom. There are different modes that can be utilized for different projects. Students can make their work public to their classmates, send a private video to a teacher, or have their own journal-type of video where only the student can view it. Beyond that, Screencastify is great for demonstrations, annotating, and mini lessons.
Cons: The free version only allows videos to be at most five minutes long.

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Overall impressions: YouTube shorts was not my favorite. It is very TikTok-esque and lends itself well to the phone screen.
Pros: Teachers can quickly write out the solution to a difficult problem, record it using YouTube Shorts, and then send that video to a student or the entire class.
Cons: It records very short clips and is not compatible with desktops.

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Overall impressions: While the site is slightly confusing, Flipgrid is great. It is extremely compatible with creating and submitting assignments and building a classroom.
Pros: The videos are versatile and the user has many options. You can add music, fun backgrounds, videos, pictures, etc. and make your video unique. This is great for presenting a class project or introductions.
Cons: The site seems to have some glitches. Upon making my introduction video, I could not get the video to start recording. Eventually, it worked out fine, and I would recommend this tool.

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Overall impressions: Nearpod is fantastic. However you teach, it's in Nearpod.
Pros: Lessons are 100% compatible with Powerpoint presentations, pdfs, etc. Additionally, teachers can either start their lesson from scratch using Nearpods built in starters, or there are thousands of pre-made lessons and activities to use or use as a base. These lessons are also created with state-standards and common core standards alike. The list for pros goes on and on.
Cons: I have no cons for Nearpod. The only downfall is if you and/or your school district do not have Nearpod premium, the base amount of storage is very small (3-4 lessons). The premium version also has a ton of perks associated with it.

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Overall impressions: EdPuzzle is a great idea, but the execution falls short. The idea is that teachers can assign videos and questions to their students either as individual tasks or whole-class assignments.
Pros: The idea is great and there are limitless videos to choose from. There are also videos that already have questions associated with them. EdPuzzle is compatible with Google Classroom.
Cons: Trish and I tried and tried to figure out how to use EdPuzzle. It is not very user-friendly. When we started to get the hang of it, we ran into other problems. Mostly the whole-class video would not work.

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Easily create beautiful interactive video lessons for your students you can integrate right into your LMS. Track students' progress with hassle-free analytics as you flip your classroom!

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Overall impressions: Kahoot is a great way for students to review quick material. The game-like feel of it brings out students' competitive sides.
Pros: It is a fun way to go over concepts like exponent rules, shapes of parent fuctions, etc. Students like Kahoot because there is a leaderboard that is often associated with candy, stickers, or bonus points.
Cons: You can't use problems that have multiple steps because many students need more time to work out a long problem. Also, some students would rather pick a question immediately for the chance at more point than actually work out a solution.

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Overall impressions: Edulastic is an awesome tool for assessment. I would definitely like to try incorporating it into my classes.
Pros: Teachers are given stats on each student, can monitor progress, adjust grades for partial credit, and more. Teachers can also choose from an assortment of pre-made questions or assignments to help build their assessment.
Cons: Like Nearpod, this is a great tool, but it gets even better with the premium version. In the premium version there is anti-cheating software and many more tools.

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Overall impressions: Peardeck is a Google Extension that lends itself well to Google Slides. It automatically displays links in the same window for convenience.
Pros: It keeps students engages. They are less likely to wander if all of the tools they need are in one presentation. Teachers can also freeze what slide students are on.
Cons: As we saw when preparing for the tech presentation at SCCTM, Peardeck may not work as well on some sites. For example, Peardeck and Nearpod seem to not be very compatible as there was a serious lag.

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Overall impressions: I prefer Quizizz over Kahoot. It is individually paced, offers more time for questions, and still has a leaderboard at the end.
Pros: Quizziz can be assigned as homework, review for an assessment, etc. The boys I tutor often do Quizziz that are between 50-60 questions longs as test review. Their teacher leaves the optional Quizziz open and allows students to go back over their work.
Cons: I have no cons. Even when students miss questions, there are tools they can use to fix their mistakes. The can freeze time, cut the answer choices in half, and even answer redemption questions to boost their score.

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Overall impressions: A product of NCTM, Illuminations is a great database filled with lesson plans, activities, brain teasers and games at every level.
Pros: Teachers can search for their grade-level, subject area, even standard and find materials for their class. They have activities from sites like Desmos and Geogebra. Their lesson plans are significant and detailed.
Cons: Not all of the activities on the site are amazing or useful, so you have to hunt around for materials that work best for you and your class.

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Illuminations Home - National Council of Teachers of Mathematics

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National Library of Virtual Manipulatives

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National Library of Virtual Manipulatives

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Overall impressions: NLVM is a simple site, but it has a lot of graphics, images, etc. that are very useful for the classroom.
Pros: The site has manipulatives from pre-k to high school. The organization of the site is user-friendly and is easy to navigate.
Cons: The site is useful and gets the job done. It is not very fancy, but I don't think it needs to be.

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National Library of Virtual Manipulatives

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A digital library containing Java applets and activities for K-12 mathematics

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Overall impressions: Like NLVM, Math Warehouse is simple but is still beneficial.
Pros: The site includes visuals, interactive activities, jokes, explanations, definitions, and worksheets! While there are some elements that are better than others, MathWarehouse is a great place where teachers can pull elements and utilize them in class.
Cons: The site itself has room for improvement, but that doesn't take away from the practicality of MathWarehouse and its functionality.

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Interactive Math Activities, Demonstrations, Lessons with definitions and examples, worksheets, Interactive Activities and other Resources

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Mathwarehouse.com--a website dedicated to Math lessons, demonstrations, interactive activities and online quizzes on all areas of geometry, algebra and trigonometry.

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Overall impressions: Photo math gives step-by-step worked out solutions to math problems.
Pros: This is a math application for mobile users. It allows students and teachers to access solutions quickly for checking their work.
Cons: This is not available on desktop computers.

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Photomath - The Ultimate Math Help App | Math Explained

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Solve even complex math problems with Photomath, the top-rated math camera solver app. Download now and understand your math homework step-by-step.

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Overall impressions: Brainly is a pretty good math application. I could see myself or students using the app.
Pros: It is available on both phones and computers. Students can scan their questions, crop the image, and the Brainly will provide the solution.
Cons: It has a lot of features that cost more money. Other applications are mostly free.

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Overall impressions: I have personally used Symbolab to check my work. It's a great mathematics application.
Pros: It has tons of symbols, variables, and operators. There is also a list of subjects that range from basic algebra to vector calculus.
Cons: There are always small flaws in these math calculators.

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\bold{\mathrm{Basic}} \bold{\alpha\beta\gamma} \bold{\mathrm{AB\Gamma}} \bold{\sin\cos} \bold{\ge\div\rightarrow} \bold{\overline{x}\space\mathbb{C}\forall} \bold{\sum\space\int\space\product} \bold{\begin{pmatrix}\square&\square\\\square&\square\end{pmatrix}} \bold{H_{2}O}

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Symbolab Math Solver - Step by Step calculator

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Overall impressions: I was very surprised and happy to find Microsoft Math Solver. It was my favorite of the math applications.
Pros: Teachers and students can share their findings on Microsoft Teams to present to the class. This application is available on both computer/web browser and on mobile devices.
Cons: I have no cons for Microsoft Math Solver.

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Microsoft Math Solver - Math Problem Solver & Calculator

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Online math solver with free step by step solutions to algebra, calculus, and other math problems. Get help on the web or with our math app.

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Overall impressions: Only available on mobile devices. yHomework left much to be desired. It has a ton of glitches, is not very user-friendly, and even has miscalculations. I would not download or recommend this application.

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yHomework - Math Solver - Apps on Google Play

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Having trouble with algebra? Equations, inequalities and graphs leave you frustrated? Tried to solve the problem but you're not sure you got it right? Meet yHomework - the math solver that actually works 100% of the time!No gimmicks, no fuss, no excuses - we simply give you the an instant full step-by-step solution and explanation.

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Overall impressions: Better than yHomework, Algebrator is available on mobile devices as well as web browser. Still, it has its flaws as well. I cannot say that I would recommend Algebrator, especially above our other math applications.

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The Addition Principle

After studying this section, you will be able to:

1. Solve equations of the form x + b = c using the addition principle.

2. Using the Addition Principle

There is a similar principle in mathematics. We can state it in words like this.

The Addition Principle

If the same number is added to both sides of an equation, the results on each side are equal in value.

We can restate it in symbols this way.

For real numbers a, b, c if a=b thenat+tc=b+ec

We can use the addition principle to solve an equation.

x + 16 + (-16) = 20 + (-16)   Use the addition principle to add -16 to both sides.

x+0=4   Simplify.

We have just found the solution of the equation. The solution is a value for the variable that makes the equation true. We then say that the value, 4, in our example, satisfies the equation. We can easily verify that 4 is a solution by substituting this value in the original equation. This step is called checking the solution.

When the same value appears on both sides of the equals sign, we call the equation an identity. Because the two sides of the equation in our check have the same value, we know that the original equation has been correctly solved. We have found the solution.

When you are trying to solve these types of equations, you notice that you must add a particular number to both sides of the equation. What is the number to choose? Look at the number that is on the same side of the equation with x, that is, the number added to x. Then think of the number that is opposite in sign. This is called the additive inverse of the number. The additive inverse of 16 is -16. The additive inverse of -3 is 3. The number to add to both sides of the equation is precisely this additive inverse.

It does not matter which side of the equation contains the variable. The x term may be on the right or left. In the next example the x term will be on the right.

17 =x+0   Simplify.

Before you add a number to both sides, you should always simplify the equation. The following example shows how combining numbers by addition separately, on both sides of the equation-simplifies the equation.

17=x+5   Simplify by adding.

17=17   ✔   It checks.

We can determine if a value is the solution to an equation by following the same steps used to check an answer. Substitute the value to be tested for the variable in the original equation. We will obtain an identity if the value is the solution.

We substitute 10 for x in the equation and see if we obtain an identity.

Thus, x = 10 is not the solution. Now we take the original equation and solve to find the solution.

-13=x-3   Simplify by adding.

Check to see if x = -10 is the solution. The value x = 10 was incorrect because of a sign error. We must be especially careful to write the correct sign for each number when solving equations.

To combine the fractions, the fractions must have common denominators. The least common denominator (LCD) of the fractions is 10.

2/10+x = 4/10  Simplify by adding.

2/10+(-2/10)+x = 4/10+(-2/10)   Add the additive inverse of 2/10 to each side

x=2/10  Add the fractions.

x= 1/5   Simplify the answer.

The Multiplication Principle

After studying this section, you will be able to:

1. Solve equations of the form 1/ax=b.

2. Solve equations of the form ax = b.

Solving Equations of the Form 1/ax=b

The addition principle allows us to add the same number to both sides of an equation. What would happen if we multiplied each side of an equation by the same number? For example, what would happen if we multiplied each side of an equation by 3?

To answer this question, let's return to our simple example of the box and the stone on a balanced seesaw. If we triple the number of weights on each side (we are multiplying each side by 3), the seesaw should still balance. The ''weight value'' of each side remains equal.

In words we can state this principle thus.

Multiplication Principle
If both sides of an equation are multiplied by the same number, the results on each
side are equal in value.

In symbols we can restate the multiplication principle this way.
|
For real numbers a,b,c wihc #0 ifa@=b thenca=cb |

Let us look at an equation where it would be helpful to multiply each side of the equation by 3.

1x=-45  Simplify.

-15=-15   ✔   It checks.

Solving Equations of the Form ax = b

We can see that using the multiplication principle to multiply each side of an equation by 1/2 is the same as dividing each side of the equation by 2. Thus, it would seem that the multiplication principle would allow us to divide each side of the equation by any nonzero real number. Is there a real-world example of this idea?

Let's return to our simple example of the box and the stone on a balanced seesaw. Suppose that we were to cut the two objects in half (so that the amount of weight of each was divided by 2). We then return the objects to the same places on the seesaw. The seesaw would still balance. The ''weight value'' of each side remains equal.

In words we can state this principle thus:

Division Principle
If both sides of an equation are divided by the same nonzero number, the results on
each side are equal in value.

Note: We put a restriction on the number by which we are dividing. We cannot divide by zero. We say that expressions like 2/0 are not defined. Thus we restrict our divisor to nonzero numbers. We can restate the division principle this way.

a b
For real numbers a, b, c where c ~ 0 ifa=b then - = -
coc

125 = 125   ✔   It checks.

For equations of the form ax = b (a number multiplied by x equals another number), we solve the equation by choosing to divide both sides by a particular number. What is the number to choose? We look at the side of the equation that contains x. We notice the number that is multiplied by x. We divide by that number. The division principle tells us that we can still have a true equation provided that we divide by that number on both sides of the equation.

The solution to an equation may be a proper fraction or an improper fraction.

(4x)/4= 38/4  Divide both sides by 4.

x=19/2  Simplify. The solution is 19/2.

If you leave the solution as a fraction, it will be easier to check that solution in the original equation.

In examples 2 and 3 we divided by the number multiplied by x (the coefficient of x). This procedure is followed regardless of whether the sign of that number is positive or negative.

(-3x)/-3=48/-3  Divide both sides by -3.

x=-16  The solution is -16.

(-1x)/-1=-24/-1  Divide both sides by -1

Use the Addition and Multiplication Principles Together

After studying this section, you will be able to:

1. Solve equations of the form ax + b =c.

2. Solve equations in which the variable appears on both sides of the equation.

3. Solve equations with parentheses.

Solving Equations of the Form ax +b=c

To solve many equations, we must use both the addition principle and the multiplication principle.

5x + 3 + (-3)= 18+ (-3)   Add -3 to both sides, using the addition principle.

5x = 15   Simplify.

(5x)/5=15/5  Divide both sides by 5, using the division principle.

Check.  18=18  ✔  It checks.

A Variable on Both Sides of the Equation

In some cases the variable appears on both sides of the equation. We would like to rewrite the equation so that all the terms containing the variable appear on one side. To do this, we apply the addition principle to the variable term.

3x = 15   Collect like terms.

(3x)/3=15/3  Divide both sides by 3.

Many problems have variable terms and constant terms on both sides of the equation. You will want to get all the variable terms on one side and all the constant terms on the other side.

9x + (-7x) + 3 = 7x + (-7x) - 2   Add -7x to both sides of the equation.

2x+3=-2   Combine like terms.

2x + 3+ (-3) = -2 + (-3)   Add -3 to both sides.

2x = -5   Simplify.

Check.   − 39/2 = − 39/2  ✔  It checks.   x = is the solution.

In our next example we will study equations that need simplifying before any other steps are taken. Where it is possible, you should first collect like terms on one or both sides of the equation. The variable terms can be collected on the right side or the left side. In this example we will collect all the x terms on the right side.

EXAMPLE 4 Solve for x.   5x + 26 -6 = 9x + 12x

5x + 20 = 21x   Combine like terms.

5x + (-5x) + 20 = 21x + (-5x)   Add -5x to both sides.

20 = 16x   Combine like terms.

20/16 =(16x)/16   Divide both sides by 16

5/4=x  (Don't forget to reduce the resulting fraction.)

All the equations we have been studying so far are called first-degree equations.
This means the variable terms are not squared (such as x^2 or y^2) or some higher power. It is possible to solve equations with x^2 and y^2 terms by the same methods we have used so far. If x^2 or y^2 terms appear, try to collect them on one side of the equation. If the squared term drops out, you may solve it as a first-degree equation using the methods discussed in this section.

5y^2-5y^2 + 6y -2 = -y + 5y^2-5y^2 + 12   Subtract 5y^2 from both sides.

6y -2=-y+12   Combine, since 5y^2-5y^2 = 0.

6y+y-2=-y+y+12   Add y to each side.

7y-2= 12   Simplify.

7y-2+2=12+2   Add 2 to each side.

7y=14   Simplify.

(7y)/7 = 14/7   Divide each side by 7.

y=2   Simplify.   The solution is2.
Solving Equations with Parentheses
The equations that you just solved are simpler versions of equations that we will now discuss. These equations contain parentheses. If the parentheses are first removed, the problems then become just like those encountered previously. We use the distributive property to remove the parentheses.

After removing the parentheses, it is important to collect like terms on each side of the equation. Do this before going on to isolate the variable.

x + 13 = 25   Collect like terms.

x+ 13-13 = 25-13   Add -13 to both sides to isolate the variable.

x = 12   The solution is 12.

4(13)-3(9) ≟ 25  Combine numbers inside parentheses.

25=25  ✔  Simplify. It checks.

In problems that involve decimals, great care should be taken. In some steps you will be multiplying decimal quantities, and in other steps you will be adding them.

0.36x-1.08 = 4.2x -16.44   Remove parentheses.

0.36x-0.36x-1.08 = 4.2x-0.36x-16.44   Subtract 0.36x from both sides.

-1.08 = 3.84x -16.44   Collect like terms.

-1.08 + 16.44 = 3.84x-16.44 + 16.44   Add 16.44 to both sides.

15.36 = 3.84x   Simplify.

15.36/3.84=(3.84x)/3.84  Divide both sides by 3.84.

6z-10 + 2 = 4z-6z-24   Remove parentheses.

6z- 8 = -2z-24   Collect like terms.

6z-8 + 2z = -2z + 2z-24   Add 2z to each side.

8z-8 = -24   Simplify.

8z-8 +8 = -24+ 8   Add 8 to each side.

8z =-16   Simplify.

(8z)/8=-16/8   Divide each side by 8.

z=-2   Simplify. The solution is -2.

Equations with Fractions

After studying this section, you will be able to:

1. Solve equations with fractions.

Solving Equations with Fractions

Equations with fractions can be rather difficult to solve. This difficulty is simply due to the extra care we usually have to use when computing with fractions. The actual equation solving procedures are the same, with fractions or without. To avoid unnecessary work, we transform the given equation with fractions to an equivalent equation that does not contain fractions. How do we do this? We multiply each side of the equation by the lowest common denominator of all the fractions contained in the equation. We then use the distributive property so that the LCD is multiplied by each term of the equation.

12(1/4x-2/3)=12(5/12x)  Multiply each side by 12

(12/1)(1/4)(x)-(12/1)(2/3)=(12/1)(5/12)(x)   Use the distributive property.

3x-8 = 5x   Simplify.

3x + (-3x)-8 = 5x + (-3x)   Add -3x to each side.

-8 = 2x   Simplify.

− 5/3 = − 5/3  ✔  It checks.

In Example 1 we multiplied each side of the equation by the LCD. It is common practice to immediately go to the second Step and multiply each term by the LCD, rather

x/7+5/7=x/4+1/2  First we write as separate fractions

4x + 20 = 7x + 14   Simplify.

4x-4x + 20 = 7x-4x + 14   Add -4x to both sides.

20 = 3x + 14   Collect like terms.

20-14=3x + 14- 14   Add -14 to both sides.

6 = 3x   Collect like terms.

If a problem contains both parentheses and fractions, it is best to remove the parentheses first. Many students find it is helpful to have a written procedure to follow in solving these more involved equations.
Procedure to Solve Linear Equations
1. Remove any parentheses.

2. If fractions exist, multiply all terms on both sides by the lowest common denominator of all the fractions.

3. Collect like terms if possible. Simplify numerical work if possible.

4. Add or subtract terms on both sides of the equation to get all terms with the variable on one side of the equation.

5. Add or subtract a value on both sides of the equation to get all terms not containing the variable on the other side of the equation.

6. Divide both sides of the equation by the coefficient of the variable.

7. Simplify the solution (if possible).

8. Check your solution.

Let's use each step in solving this example.

EXAMPLE 3 Solve for x and check your solution. 1/3(x-2)= 1/5(x+4)+2

Step 1   x/3-2/3=x/5+4/5+2  Remove parentheses.

Step 3   5x-10 = 3x + 12 + 30   Simplify.

5x-10 = 3x + 42   Simplify.

2x-10 = 42   Simplify.

It should be remembered that not every step will be needed in each problem. You can combine some steps as well, as long as you are consistently obtaining the correct solution. However, you are encouraged to write out every step as a way of helping you to avoid careless errors.

It is important to remember that when we write decimals these numbers are really fractions written in a special way. Thus, 0.3 = 7 and 0.07 = 745. It is possible to take a linear equation containing decimals and to multiply each term by the appropriate value to obtain integer coefficients.

Formulas

After studying this section, you will be able to:

1. Solve formulas for a specified variable.
Solving for a Specified Variable in a Formula
d= rt   Use the distance formula.

156 = 52t   Substitute the known values for the variables.

156/52=52/52t  Divide both sides of the equation by 52 to solve for t.

3=t   We have solved for t.

It took Joseph 3 hours to drive 156 miles at 52 miles per hour.

If we have many problems that ask us to find the time given the distance and rate, it may be worthwhile to rewrite the formula in terms of time.

d/r=t   You have solved for the variable indicated.

A simple first degree equation with two variables can be thought of as the equation of a line. It is often useful to solve for y in order to make graphing the line easier.

(-2y)/(-2)= (6-3x)/(-2)  Divide both sides by the coefficient of y.

y=6/-2+(-3x)/-2  Rewrite the fraction.

y= 3/2x-3   Simplify and regroup.

This is known as the slope-intercept form of the equation of a line.

Our procedure for solving a first-degree equation can be rewritten to give us a procedure for solving a formula for a specified variable.
Procedure to Solve a Formula for a Specified Variable
1. Remove any parentheses.

2. If fractions exist, multiply all terms on both sides by the LCD of all the fractions.

3. Collect like terms or simplify if possible.

4. Add or subtract terms on both sides of the equation to get all terms with the desired variable on one side of the equation.

5. Add or subtract the appropriate quantity to get all terms that do not have the desired variable on the other side of the equation.

6. Divide both sides of the equation by the coefficient of the desired variable.

7. Simplify if possible.

EXAMPLE 4 A trapezoid is a four-sided figure with two parallel sides. If the parallel sides are a and b and the altitude is h, the area is given by

Solve this equation for a.

A=(ha)/2+(hb)/2  Remove the parentheses.

2(A) = 2((ha)/2)+2((hb)/2)   Multiply all terms by LCD of 2.

2A = ha + hb   Simplify.

(2A-hb)/h= (ha)/h  Divide both sides by h (the coefficient of a).

(2A-hb)/h=a  The solution is obtained.

Note: Although the solution is in simple form, it could be written in an alternative way. Since

(2A-hb)/h=(2A)/h-(hb)/h=(2A)/h-b

we could have (2A)/h-b = a as an alternative way of writing the answer.

Write and Graph Inequalities

After studying this section, you will be able to:

1. Interpret an inequality statement.

2. Graph an inequality on a number line.
Inequality Statements
We frequently speak of one value being greater than or less than another value. We say that ''5 is less than 7'' or ''9 is greater than 4.'' These relationships are called inequalities. We can write inequalities in mathematics by using symbols. We use the symbol < to represent the words ''is less than.'' We use the symbol > to represent the words ''is greater than.''

Statement in Words        Statement in Algebra

Note. ''5 is less than 7'' and ''7 is greater than 5'' have the same meaning. Similarly, 5 <7 and 7 > 5 have the same meaning. They represent two equivalent ways of describing the same relationship between the two numbers 5 and 7.

We can illustrate the concept of inequality graphically if we examine a number line.

+++ +-_ +++ +_+_+-_-_+_¢_ _ +>
-5 -4 -3 --2 -] 0 I 2 3 4 5 6 7 8

EXAMPLE 1 Replace the question mark with the symbol < or > in each statement.
Graphing an Inequality on a Number Line
-5 -4 -3 -2 -1 0 l 2 3 4 5

Note that the open circle at 3 suggests that we do not include the point for the number 3.

Similarly, we could represent graphically x < -2 as follows:

$$ -} fj -_ + -_;_+_+_+_ + > x
-5 -4 -3 -2 -l 0 I 2 3 #4 = =# §

-+-. >! 4H_{_AH_ ee ee et
-2 -1 0 l 2 3 4 5 6 7

Note that the closed circle at 3 suggests that we do include the point for the number 3.

Similarly, we could represent graphically x <= -2 as follows:

-- t+. et Ht HH HH
-5 -4 -3 -2 -1 0 1 2 3 4 5

EXAMPLE 2 State each mathematical relationship in words and then illustrate it graphically.

(a) We state that ''x is less than -2."

x<-20 --+-+-O-_1+-_+-_+-_++-
-4 -3 -2 -!1 0 ] 2

(c) We state that ''x is less than or equal to -6."

_--od-_tH¥!_t-_t*-_+-_ +t +
-7 -6 -5 -4 -3 -2 -1 0 l 2

There are many everyday situations involving an unknown value and an inequality. We can translate these situations into algebraic statements. This is the first step in solving word problems using inequalities.

EXAMPLE 3 Translate each English statement into an algebraic statement.

(a) The police on the scene said that the car was traveling greater than 80 miles per hour (use the variable s for speed).

(b) The owner of the trucking company said that the payload of a truck must never exceed 4500 pounds (use the variable p for payload).

(a) Since the speed must be greater than 80 we have s > 80.

(b) If the payload of the truck can never exceed 4500 pounds, then the payload must be always less than or equal to 4500 pounds. Thus we write p <= 4500.

Solve Inequalities

After studying this section, you will be able to:

1. Solve an inequality.
Solving Inequalities
The possible values that make an inequality true are called its solutions. Thus, when we solve an inequality, we are finding all the values that make it true. To solve an inequality, we simplify it to the point where we can clearly see the possible values for the variable. We've solved equations by adding, subtracting, multiplying, and dividing a particular value on both sides of the equation. Here we do similar operations with inequalities, with one important exception. We'll show some examples so that you can see the operations we can do with inequalities just as with equations.

We will first examine the pattern that takes place when we perform a given operation on both sides of an inequality.
EXAMPLE 1
Original Inequality                          New Inequality

Note that we avoided multiplying or dividing by a negative number !

Now let us examine what would happen if we did multiply or divide by a negative number. We start with an original, true inequality. We want to get a new, also true inequality.

Original Inequality                    New Inequality

The < sign we started with (3 < 5) is reversed to > (-6 >-10). A similar reversal takes place in the following example.
EXAMPLE 2
Original Inequality                          New Inequality

Notice that we do the arithmetic with signed numbers just as we always do. But the new inequality has its inequality sign reversed (from that of the original inequality). Whenever both sides of an inequality are multiplied or divided by a negative quantity, the direction of the inequality is reversed.

Procedure for Solving Inequalities
| You may use the same procedures to solve inequalities that you did to solve equa-
tions except that the direction of an inequality is reversed if you multiply or divide
both sides by a negative number.

EXAMPLE 3 Solve and graph 3x + 7 >= 13.

3x +7-7>=13-7   Subtract 7 from both sides.

3x>= 6   Simplify.

(3x)/3>=6/3   Divide both sides by 3.

x>=2   Simplify. Note the direction of the inequality is not changed, since we have divided by a positive number.

The graphical representation is en ee
-2 -Il 0 I 2 3 4

EXAMPLE 4 Solve and graph 5- 3x > 7.

5-5-3x>7-5   Subtract 5 from both sides.

-3x>2   Simplify.

(-3x)/-3<2/-3  Divide by -3, and reverse the inequality, since both sides are divided by negative 3.

x< -2/3   Note the direction of the inequality.

The graphical representation is Ht Ht
-1_2_1 0 l
3 3

Just like equations, some inequalities contain parentheses and fractions. The initial steps to solve these inequalities will be the same as those used to solve equations with parentheses and fractions. When the variable appears on both sides of the inequality, it is advisable to collect the x terms on the left side of the inequality symbol.

EXAMPLE 5 Solve and graph −(13x)/2<=x/2-15/8

(8)((-13x)/2)<=(8)(x/2)-(8)(15/8)  Multiply all terms by LCD = 8. We do not reverse the direction of the inequality symbol since we are multiplying by a positive number, 8.

-52x <= 4x-5   Simplify.

-52x-4x <= 4x-15-4x   Add -4x to both sides.

-56x <= -15   Combine like terms.

(-56x)/56>= -15/-56  Divide both sides by -56. We reverse the direction of the inequality when we divide both sides by a negative number.

The graphical representation is 0 15 28 l
56 56
. ]

The most common error students make in solving inequalities is forgetting to reverse the direction of the inequality symbol when multiplying or dividing by a negative number.

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solving equations
math equations
linear equations
solving inequalities
algebra equation
graphical equation solution

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After studying this section, you will be able to: 1. Solve equations of the form x + b = c using the addition principle. 2. Using the Addition Principle   There is a similar principle in mathematics. We can state it in words like this.

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Overall impressions: Remind is easy to use and effective. Parents and students receive notifications about assignments, tutoring, and homework.
Pros: Teachers can easily send mass text messages out to remind their students and parents about assignments.
Cons: There are boundaries when it comes to subscriptions and payments.

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Overall impressions: Slack is a great tool to facilitate conversations in the classroom. It also works well with professional networks.
Pros: Slack has options to set up multiple channels with administrators, moderators, etc.
Cons: It may be difficult to monitor from a school district perspective.

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Overall impressions: GroupMe is a group chat application that I am quite familiar with. I do not know if it lends itself well to the school platform since it is hard to monitor. They currently do not have any way to monitor chats. There is a way to remove people, but even that does not seem appropriate for the students in a class setting.

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They say more than a sentence ever could. Snap a pic on the go or share an old one to relive some memories together.

Share the sights and sounds of your life as it happens. Take a video right from the app, or upload an old memory to share with your friends.

Easily explore all the photos that have been shared to every group you are in.

Sometimes a message doesn't need a text response. Tap the heart to show you care.

Who knows the address of anything anymore? Send where you are or where you're going to be so no one gets lost.

Say more with our expressive emoji. These little guys will make your conversations cuter and much more fun, and we release new ones all the time!

Not every message should go to the whole group. Tap a friend's avatar anywhere in the app and take your private conversation aside.

Some groups can get pretty chatty. Mute them for a little peace. Or, if you have a meeting, easily mute everything for a while.

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GroupMe | Group text messaging with GroupMe

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They say more than a sentence ever could. Snap a pic on the go or share an old one to relive some memories together. Share the sights and sounds of your life as it happens. Take a video right from the app, or upload an old memory to share with your friends.

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Function Builder on PhET is a great tool for students when exploring functions. This can be used in the classroom when students are ready to advance from symbols/words in functions and relate it back to algebra. This also displays a table so students can keep track of the changes.

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Play with functions while you ponder Art History. Explore geometric transformations and transform your thinking about linear functions, then have fun figuring out the mystery functions!

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The predecessor to Function Builder, Function Builder: Basics is a great way to introduce functions at any age. This one is neat because it includes motions and colors, so students can take note about what is being done to an image. A conversation about symmetry can also arise here since some images will appear to not have changed.

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Play with functions while you ponder Art History. Look for patterns, then apply what you learned on the Mystery screen!

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Number Line: Operations on PhET is an impactful visual tool when it comes to integer operations. Seventh grader specifically could benefit greatly from this tool. The color-coded aspect of this tool is my personal favorite.

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Explore operations with integers using an authentic context of net worth, then generalize with a number line. Experiment with adding and subtracting both positives and negatives, and make predictions about whether the total will be positive or negative.

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Matching fractions is typically a difficult task for younger students. This application provides visual representations that allow students to see the same fraction written in different ways. PhET also gamified this app, so students are awarded with points if they get the correct answer.

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Match shapes and numbers to earn stars in this fractions game. Challenge yourself on any level you like. Try to collect lots of stars!

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Arithmetic is a tool on PhET that would best be used in a 3rd-4th grade class learning their multiplication tables. Rather than pure memorization, students are able to make connections and reason for themselves using this application.

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Remember your multiplication tables? Practice your multiplication, division, and factoring skills with this exciting game. No calculators allowed!

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This is my favorite of all of the PhET simulations we observed. This tool allows students to work with and explore graphing using slope intercept form. The form y=mx+b is emphasized, the points are color coded, and students can use the point-finder to show what the coordinate is.

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Explore the world of lines. Investigate the relationships between linear equations, slope, and graphs of lines. Challenge yourself in the line game!

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Equality Explorer on PhET allows students to explore equations by making the scale even. This is great since the green arrow shows when the scale is even and the action performed is displayed above the two sides of the scale. Students can use this tool and write down their steps or observations.

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Explore what it means for a mathematical statement to be balanced or unbalanced by interacting with objects on a balance. Discover the rules for keeping it balanced. Collect stars by playing the game!

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Similar to Graphing Lines, this is a great PhET simulation. Students are encouraged to experiment by changing the slope and/or slope intercept, finding points on the line, and saving lines to compare against other lines.

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Explore slope-intercept form of a line. Connect the slope and y-intercept to the equation of the line. Challenge yourself in the line game!

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This is an example of Screencastify. I am demonstrating how to set up a Screencastify assignment that is compatible with Google Classroom. This assignment allowed students to submit a video with questions that the teacher could privately answer.

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I significantly edited a premade lesson I found on Desmos. Desmos Activity builder allowed me to mold the lesson to fit my needs and my students needs.

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In this delightful and challenging activity, students will transform lines so that the marbles go through the stars. Students will test their ideas by launching the marbles and will have a chance to revise before trying the next challenge.

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3 Act Math Lesson Plan, Mathematical Modeling

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3 Act Math Lesson Plan, Mathematical Modeling

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This was a lesson plan from one of my least favorite lessons. While my lesson plan is extremely detailed, I was not prepared for the lesson. I was still uncomfortable with the textbook software/video and did not know how to engage my students.

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Here is an example of Polygraph. It is a Guess-Who like game that lends itself well to the classroom. Students ask each other questions to narrow down the graphs to the correct one. It's suitable for all ages.

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I found this site when I was building my Bitmoji classroom. I can really see myself going back to it when I have a classroom of my own and start incorporating weekly (or daily) jokes.

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What if instead of groans, when you said it was time for math, students cheered?

Unfortunately, math has gotten a bad rap for being boring and difficult. But, there's a lot that math teachers can do to make math funny and enjoyable. Math jokes and math puns are one of the best tools teachers have to bring a lighter side to the math classroom.

Are you ready to giggle, laugh and chuckle at some math jokes? Below read 30 of the funniest math jokes for teachers and then learn how to use them in the classroom to get the best outcomes:
Funny Math Jokes
1. Hey have you ever noticed what's odd? Every other number!
2. Do you know what mathematicians do after it snows? They make snow angles!
3. Which tool is best for math? The multi-pliers.
4. Swimmers love one kind of math more than all others, what is it? Dive-ision!
5. Once there was a hen who counted her own eggs. She was a mathemachicken!
6. Why was six afraid of seven? Because seven, eight, nine!
7. Did you know that there are three kinds of people in the world? Here they are: people who can count and people who can't.
8. Do you know why the two 4s didn't go to the cafeteria for lunch? They already 8!
9. Do you know the special trick to turn seven into an even number? Remove the s!
10. What do the moon and a dollar have in common? They both have 4 quarters.
11. What don't atheists do well with exponents? Because they don't believe in higher powers.
12. The English book asked the Math book why he was so sad. You know what he said? "Because I have so many problems!"
13. Why is 69 so scared of 70? Because once they fought, and 71.
14. Who's in charge in a pencil case? The ruler.
15. Why can't your nose grow to be 12 inches long? Because then it would be a foot.
16. Which numbers just won't sit still? Roamin' numbers
17. Did you hear what the 0 said to the 8? Nice belt!
18. Do you know the trick for making time fly? Throw a clock out a window!
19. If you're cold, go to the corner of the room. It's 90 degrees there!
20. Why did the girl always wear glasses when doing math? It improved her division!
21. The minus sign was talking to the positive sign. The minus sign asked, "Are you sure I make a difference?" and the other sign said "I'm positive!"
22. A teacher asked her student "Why are you doing math on the floor?" The student answered, "You told us not to use any tables!"
23. There was once a talking sheepdog. He ran up to the farmer and said "All 70 sheep are in the pen."
The farmer said, "But I only counted 67!"
The sheepdog said "Yeah, but I rounded them up!"
24. What's the best way to get a math tutor? An add!
Geometry Jokes
25. Why did the circle get offended by the triangle? Because the triangle looked at the circle and said "You're pointless!"
26. Do you see that circle over there? He's really smart. He has 360 degrees.
27. Do you know which tree is the math teacher's favorite?
Geometry!
28. It's so sad to think that parallel lines have so much in common...but they'll never be able to meet!
29. Why shouldn't you ever argue with a 90 degree angle? They're always right!
30. Why are obtuse angles always so sad? They're never right!
Algebra Jokes
31. Do you know what type of math birds like best? Owlgebra!
32. Do you know who invented algebra? An x-pert.
33. Which civilization was best at algebra? The Romans. For them, X was always 10.
34. Algebra can make you a better dancer. Do you know why? Because you can use an algo-rhythm.
35. What do you call a group of friends who love math? AlgeBROS.
How to Make Math Funny
How can you use corny math jokes to make math class fun and interesting? You don't have to recite math jokes all class to make it fun. Here are some creative ways to use math jokes for kids:

Math Joke of the Day
Start math class by writing out part of a math joke on your board. Ask kids to try to think of the answer. Then, reveal the answer to your students when you start class!

Funny Math Puns
Many numbers and math words have two meanings and can be used to make puns. For example "Why was six afraid of seven? Because 7,8,9." The pun is on the word "eight", which can also mean "ate".

Tell a few puns to your students and invite them to think of their own puns using math words with double meanings such as:
Four and for
Two and to
Eight and ate
Nine and nein (the German word for "no")
Acute and cute
Pi and pie
Sign and sine
Add and ad
Math Problem Jokes
You can also use the best math jokes to motivate kids as they work through math problems. List the start of the joke on the worksheet. Then, to get the answer, students have to work through the problems.

For example, make a worksheet in which the answers to the problems each represent a letter. Then, when they've finished all of the problems, they can either unscramble the letters or you can provide an answer key to help them.

Are you ready to use math jokes in your classroom?

Tell us how you use jokes to make math fun in your classroom. You can't ever have 2 much fun with math!

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30 Funniest Jokes for Math Teachers - Math Blog for Differentiation

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What if instead of groans, when you said it was time for math, students cheered? Unfortunately, math has gotten a bad rap for being boring and difficult. But, there's a lot that math teachers can do to make math funny and enjoyable.

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Converting Forms of Linear Equations, Nov 2021

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Converting Forms of Linear Equations, Nov 2021

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This is my recorded lesson, Converting Forms of Linear Equations. This was recorded at the very end of my Int A2. I was familiar with my students and their needs. This lesson was given two days before their Unit 5 test and their last day of school before Thanksgiving break.

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Diversity and Equity in Ultimate: Honors Thesis, DeLuca (Aug. 2021)

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Diversity and Equity in Ultimate: Honors Thesis, DeLuca (Aug. 2021)

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Finally, here is a pdf copy of my Honors Thesis. I wrote about diversity and equity in the ultimate frisbee world, specifically college. The governing body, USA Ultimate, has recent updated and rewritten their EDI policies. I conducted survey research and explored email communications. This all culminated into a series of suggestions for USAU to implement.

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While we did not go over this simulation in class, Ratio and Proportion is a two-part PhET activity. In the first part, students can discover the relationships between ratios and proportions. In the second part, they can create their own ratios and proportions. This would be a great tool in a 5th-6th grade classroom.

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This was my Geogebra activity. I modeled this lesson on one I did over the summer. It was an exploration/discovery activity on the unit circle. We start by recalling special right triangles and then using those patterns to figure out the unit circle.