###### September 24, 2023

### What’s The Big Deal About Math?

**What really makes this educational discipline so integral to our learning as humans?**

Math, as a whole, has become such an intrinsic part of comprehensive education.

Without the calculated use of numbers, shapes, equations, and so much more- we wouldn’t have made nearly as many advancements we use in our daily lives today! Isn’t that crazy?

Although math is a subject many people have a hard time grasping, it’s greatly applicable to assure our general wellbeing. Those who are more adept in the field have the power to alter the course of our evolution- FOREVER. And even those who don’t choose to make a career out of it can still possess applicable life skills to make their daily living that much easier.

**Math was originally discovered and used to construct empires and their systems of living in ancient times.** Nowadays, it’s applied to nearly everything from cooking to architecture to dance choreography. We measure and calculate and articulate so many things in our habitual lifestyle it’s easy to overlook how much math we exercise consistently.

#### What makes math SO important, you ask?

Well, here are six core reasons why Math is incredibly important to include in a diversified model of education:

**Math promotes critical thinking skills**: Critical thinking is an important skill covering a breadth of other cognitive abilities such as analysis, interpretation, inference, explanation, self-regulation, open-mindedness, and problem-solving. Math utilizes its kind of critical thinking to get students (or anyone using it) to brainstorm reasoned decisions to approach particular problems. Rather than guessing or applying a rule without testing its validity, they have to be able to concoct educated judgments that best suit the necessary solution to the problem.**Truth be told, STEM is the future**: STEM includes varying math-oriented disciplines such as science, engineering, and technology. These are fields able to help our technological evolution progress, so it’s incredibly important to implement the fundamental learnings that provide the solid basis for pursuing and studying such prominent endeavors.**Logical thinking is a communicable life skill**: Unlike critical thinking, logical thinking is how to evaluate the processes behind the reasoning. Critical thinking uses logic and its core principles to separate truth from falsehood. However, it doesn’t focus on said reasoning specifically. Solidifying a strong math education, especially for those at a young age, can empower them to make creative yet careful decisions.**Math productively teaches about failure**: Look, we’ve all experienced an excruciatingly difficult math problem we SWORE we would be unable to solve. But sure enough, that’s one of the most crucial lessons math teaches us- failure is the opportunity to learn even more. When someone fails to solve a problem, they have the option to give up or attempt the problem again. If they choose to try again, this strengthens their learning process; if not, then they’re stuck in a limiting belief they can’t achieve a resolution. Math subtly emphasizes that failure is a part of reality, and we have the choice to turn it into something greater.**Math ALSO exemplifies the importance of a growth mindset**: Math has the power to teach us that competence is an ever-evolving process- NOT a limited set of skills. Combining a growth mindset with the reality of failure prepares young students to attack their decision-making with patience and an awareness that they have the power to reach their end goal! These are two very essential skills in optimal social, emotional, and cognitive development.**It helps strengthen financial literacy skills for the future**: There comes a day where many of us are given the choice to inhabit the responsibility of supporting ourselves (and in some cases, others). Math is so deeply ingrained into financial processes, it’s unavoidable when taking on those monetary tasks. If you assess and incorporate strong math education at an early age, financial literacy is much more likely to be retained at an easier and faster rate!

Mathematics is so deeply incorporated into our daily functions, and the mechanic behind our most used tools, that it’s nearly impossible to avoid. Our societal standard of success is based greatly upon our inherent ability to make informed, sound, logical decisions; we don’t even realize how much math helps us to condition this technique!

If you encounter a young student with a difficulty in learning math, now is the time to encourage these key principles. Math can be taught in environments outside of the traditional classroom, such as the kitchen or certain extracurriculars. Although that young child may not NEED math for their future professional endeavors, it’s still going to be a functional skill once they reach adulthood.

Even though math is a meticulous discipline, it has the power to better our means of thriving. And without it, who knows where we’d all be!

###### September 17, 2023

### Let’s Go Over This Again, Shall We?

**Why Review in Math Education is ESSENTIAL for Student Retention- and Good Grades**

It’s very common among educational institutions that some students will present a much more difficult time retaining learning. Whether it be a learning divergency, inability to pay attention, or the content itself is too difficult to grasp- many students all over the world struggle with education retention.

It isn’t entirely their fault. Young minds are so malleable they all have various ways of learning new content. Memory is an especially intricate function of the human brain; it only makes sense that a young child’s ability to remember is slightly warped due to their lack of biological maturity as well as experience. They take in SO much information throughout a day, it’s only rational to understand the overwhelm they must feel! According to an article by Neuron, forgetting is “a critical component of a healthy mnemonic system”.

However, the instructors and teachers in place have a responsibility to best try and coordinate different modalities for learning to ensure every student is receiving what they need.

**What To Do About Forgetfulness in Education**

A very traditional yet consistently successful way to increase learning retention is constant, even meticulous review. This has proven to be especially beneficial when it comes to teaching young students Math. Math is one of the more objectively logical subjects in school, with a multitude of tiny moving parts and rules.

Because math is a particularly dense yet detailed subject, with LOTS of steps to completing certain problems, children have a hard time keeping it all fresh and at their disposal.

Extended breaks typically diminish the potency of learned knowledge, which makes review all that more beneficial. To review means to look back at all the lessons taught, read through, maybe even practice problems again and again for optimal comprehension.

Math has been proven a subject that strongly implicates the importance of a growth mindset. This is especially representative through the learnedness of the review!

In math, a student is given the option to attack a failed problem or equation two ways: try again with a new approach, or disregard it entirely. How does this tie into Math review?

If a student (given the proper resources) REVIEWS the mathematical fundamentals they’re currently learning, they’ll eventually discover the flaws in their problem solving, thus being more motivated to try again with a more developed solution. Converging review with a growth mindset encourages the students to practice patience, resilience, and a more optimistic outlook.

Additionally, the review is representative of a growth mindset because it encourages students to look outside themselves for resources (when available).

The vital aspect of the review is to be able to connect the content. If a student has an upcoming assessment with more than a few mathematical techniques being addressed, then they must be provided the insight into how they’re all related. The more the students are assisted in recognizing the interconnectedness of math, the more retention they constitute.

**Fun Ways to Implement Math Review**

Yes, there are FUN and exciting ways to integrate math review into a student’s coursework. There is a multitude of approaches, ranging from collaborative work to games even to additional class resources!

Here are just some (of the many) ways to enjoyable implement math review- in a classroom AND at home!

**Games**: Everybody loves a good game! Math is not everyone’s favorite subject to learn at school, so incorporating fun into the learning process makes it all that more digestible. These types of reviews can vary based on inspiration, like Math Jeopardy or Math Candyland where each student has to complete a problem to advance on the board. A Math trivia is also a great option- Kahoot has become a very fun option for classroom review!

**Collaborative Work + Peer Review**: It’s amazing what kids can learn from one another. Throughout a unit of lessons (that’ll be included on the same assessment), continuously include time for peer work on classwork problems so they can go through them TOGETHER. This is also a wonderful opportunity to strengthen their collaborative teamwork skills!

**Extra Help Opportunities**: Sometimes, the best resource is the teacher. If you are an instructor of any sort, try to allot time for students to have the option for extra help. Not only will this provide wonderful one-on-one focus to address the specific obstacles of each student, but it’ll strengthen trust in the class overall!

**Study Guides**: Depending on the age of the student, take-home study guides can work wonders! Teachers can subliminally add questions they plan to include on a test, especially the more difficult ones. Study guides are a great, direct framework for what you expect the students to know and complete!

**Access to Classroom Resources**: This can also vary depending upon the classroom. For some students, they learn best when they can look back at the notes and reread what they were taught. If the students are at an appropriate age to access work via computers, create a page entirely dedicated to your Powerpoints (or in-class exercises) as a review option!

**Computer Games**: This is a GREAT At-Home tool! Kids of many ages and levels of math difficulty can do this. There is an assortment of computer games online that a child can access to review what they’re learning. If you know your students have utilized Kahoot for review, this is something they can access at home (given computer access). Additionally, smaller and younger children who enjoy games will probably experience less resistance to math learning if there is fun included.

**An Important Factor To Consider**

As recently stated, there are going to be students who don’t receive Math well if they don’t particularly feel passionate about the subject. That’s why in any learning or review capacity it’s important to incorporate Math problems students will ENJOY. Regardless of what review is being implemented, students typically respond better when they can relate to what’s being given. This can range from (consensually) including students’ names in word problems, adding their favorite characters to problems, or finding acronyms and fun nicknames to describe mathematical concepts.

###### September 3, 2023

### What Makes A GREAT Math Tutor?

**What are the qualities & skills to look for when searching for an effective math tutor for your student?**

As many of us know, every student is different when it comes to learning. It’s no secret that not every student can retain or learn math easily and enjoyably. That’s quite alright! Unfortunately, it’s still something they must encounter throughout their schooling. What’s a kid to do?

To minimize the challenges that math presents, it’s a great idea to seek a potential resource outside of the child’s traditional classroom. There are various tutors available, both in-person and virtually, able to teach math in new or different ways than conventional school teachers. Not only are they outside their typical learning environment, but it’s a much more individualized and intimate setting that will give the child the attention they need to tackle their math struggles.

What should you exactly look for in a tutor? An effective and experienced tutor will encompass a wide array of useful skills and qualities to establish a worthwhile experience for the student.

Below are just a handful (of the ever-expanding) qualities of a GREAT math tutor.

**Qualities + Skills Of A Great Math Tutor**

**Great, But Growing, Knowledge of Math**: It’s important to give a young student a tutor that knows what they’re teaching. Having a prominent amount of mathematical knowledge is one of the most fundamental traits when looking for a good tutor. Not only should they have a foundational knowledge of the subject, but they should be open to expanding their perspective as math evolves with the times.

**Humility**: A humble, grounded tutor will provide an environment for your students that’s in THEIR best interest. The student is there to receive encouraging guidance on how to tackle math, not to boost the ego of the tutor. That’s why a great tutor should recognize their skills as worthwhile and useful, but utilize them for the good of others!

**Ability to Individualize Strategies for ALL Students**: Not every student is the same, especially when it comes to HOW they retain dense or difficult knowledge. Math, although very logical and quantitative, cannot be taught to everybody the same way. An experienced math tutor should take the time and care to assess the individual learning styles of every student they tutor to best plan their approach to teaching math at hand.

**Accountability**: There is a certain level of responsibility a great tutor must be able to take on. They have a wide assortment of students to educate, a schedule to maintain & update, and even families to interact with to communicate student progress. Accepting this role with an ethical and honest attitude ensures they will be able to address the unique circumstances of each tutoring relationship in a way it wholly deserves.

**Enthusiasm for Learning + Growing Math Skills**: How can you expect a young child to not get excited to learn about math if the tutor can’t be? Showing up to the tutoring session with a passion will motivate the student to TRY; the contagiousness of the enthusiasm may even get them excited to learn. An excellent math tutor will attack their job with fervor to help in any way they can, and even be excited to learn more about math themselves!

**Flexibility**: There will be times when a last-minute cancellation or rescheduling occurs, or maybe you’re a virtual tutor tackling technical obstacles. Regardless of the challenge, a great math tutor will maintain a flexible yet understanding mindset! The accessibility of a student will vary, so the tutor themselves should take as much responsibility as they can to do their part to ensure the students receive the resources they need.

**Professionalism**: In reality, the tutor isn’t just interacting with the student! There are probably guardians, parents, even fellow educators who want to be kept in the loop on the student’s progress; that can be overwhelming! An experienced tutor knows that providing an approachable, personable, and well-composed attitude so the dynamic remains beneficial and positive!

**Patience AND Positivity**: Learning is a marathon, not a sprint. Students will only learn as far as their motivation, focus, and mood can allow in various sessions. That’s why an exceptional math tutor will be patient yet challenge the student within certain boundaries. Not only that but maintaining positivity that the student will succeed in due time will rub off on them; children tend to energetically absorb what’s occurring around them. If you want the student to have a positive attitude, lead by example!

**Able to TEACH, not just lecture**: To fully retain or understand a mathematical concept, a student may need an in-depth explanation about a specific equation or concept. More likely than not, they won’t comprehend the math well if they’re just lectured to. It’s up to the math tutor to engage in discourse that fully encompasses the use, function, and steps of a math problem. Additionally, not many people learn effectively by just being talked to; they need various forms of education for their brain to absorb what’s being taught.

**Motivation**: Evidently, there will be sessions and/or weeks where the students will just not have the same determination to do the work they usually do. Despite the challenges, especially an energetic ebb and flow, a wonderful math tutor will persevere by brainstorming new yet engaging ways to teach; they will also know when to dial back on the studying to energetically match the student in the session. As long as the tutor has the drive to do the best they can to teach to the specific individual, it’ll be worthwhile.

**Reliability + Preparation**: Being a good math tutor doesn’t just mean showing up. It’s about being able to facilitate a productive session about the content the student needs to learn. That comes with showing up consistently, putting in your all, and most importantly coming PREPARED. A good math tutor will prepare for a session by outlining the lesson and its concepts whilst providing lots of exercises and reviews to effectively conceptualize the lesson.

###### August 13, 2023

### Math Fun

**Have Your Fun, and Learn From It Too! — Why Games Can Make for a GREAT Option to Optimize Math Review**

Let’s Have Some FUN

To sharpen learning retention in kids, it’s highly advantageous to have their full attention. And it’s even more advantageous to express the learning review in a way that’s digestible yet productive.

This is especially applicable and prevalent in math education, a discipline that requires a lot of practice and logistical awareness for the functionality of the techniques being taught. And hey, that can be unrealistic when you’re constantly lecturing them. Talking AT the students about math, or whatever they’re learning, doesn’t fully communicate the knowledge they need in order to understand what they’re being told. And even when you work one-on-one, instilling a two way path of communication about math, it may feel like pulling teeth with some kids. Admittedly, it isn’t the most enthralling of subjects for some young minds, and their resistance shows.

Another concern regarding communicable math review is the style in which you provide it. When addressing a class collectively, you’re catering to all kinds of learning styles. This can mean you have a cultivation of visual, auditory, kinesthetic, and even audio-visual learners. And although they each have individualized methods of retaining information, it is your job as an educator (or guardian) to mold a comprehensive yet constructive way to deliver the knowledge you need to.

Imagine being able to provide your students a tool, modality, or resource that not only TEACHES them the math they need, but grabs their attention in a positive and engaging way. Wouldn’t that be just AWESOME?

**What’s the solution, you ask?** Certainly NOT dumping a heavy review on them with no direction or energetic buffer. Instead, a better approach may be to keep it lighter in deliverance!

It’s become increasingly popular to include more lighthearted, enjoyable math reviews in the classroom through games. Yep, that’s right- games! Board games, computer games, CARD GAMES… that enjoyable pastime we highly encourage kids to participate in. And because it receives a moderately positive response from kids, it’s a prime candidate for utilization in school!

Why are they so beneficial you ask? Well, the primary appeal of games is that they’re designed to be lightly competitive but enjoyable. Not only that, they’re also easy to grasp once you play a few times (at least kid-friendly games are). If you were to find a way to incorporate math into a widely-known game structure, you create a refreshing review that actually gets the students excited to try it!

And what students love most about games is the childlike creativity or fun! If you bring a recognizable character, storyline, even movie into your classroom and use it to communicate your teaching, the students may grasp it more attentively! They naturally gravitate towards things they find enjoyable.

**What games are best to use in a classroom?** Well, that’s entirely situational.

If you have a smaller class size, you can split the class evenly and do multiple rounds of a one-on-one style game. For the larger-sized classes, team games are always a great way to instill community, teamwork, accountability, and enthusiasm! Students always receive teams well because they don’t want to feel isolated in their learning, and a few minds are always better than just one.

Ask yourself about these important factors when creating math review games:

Is this a game that is familiar to my students, or at least digestible in terms of difficulty?

If I give them the clear-cut directions and allow a few practice rounds, will they be able to effectively retain the knowledge while participating in the game?

Does this game have any offensive, discriminatory, or inappopriate content originally included that I can take out for my purposes?

Does this game ensure I can implement the exact math I’m teaching relatively easily?

What are some clear cut boundaries I need to set with my students?

After careful consideration, outline a game that you believe will deliver what you’re aiming to.

And if you’re looking for some guidance, look no further…

**Games for Inspiration**

**Trivia/Jeopardy**: Split the class into a few medium sized teams and create trivia style questions about math. You can give them whiteboards or paper to work out the problem on the screen, all the while collaborating with their classmates. Whoever has the correct answer first wins whatever reward was assigned to the question! This also broadens their knowledge of the discipline overall. A popular electronic trivia game is Kahoot!

**What’s Missing**: Memory is such an integral skill in math, and it should be exercised to the best of its ability! Present an equation, or visual representation of a problem (this is a great game for geometry-based questions) and ask the students what step or factor is missing! This helps them increase awareness for problem solving as well as formula-based questions.

**True or False**: Present a math equation or world problem (bonus if it’s on your assessments) and give the students an answer. It’s up to them to work on the problem on their own and decide if the answer you originally gave them was the correct one or a false one. This is also a great opportunity to work through the problem as a class AFTER you tell them if it’s true or false; some students may get it wrong, so if you show them the mistakes or misconceptions they’ll retain it. And this also helps them create a growth mindset!

**Match the Equation**: Matching card games are super quick, easy, and fun! You can create a deck that splits into: problems vs. answers, formulas vs. what they’re meant for, shapes vs. dimensions (size, shape, sides, etc). This is a flexible game style that’s suitable for all ages!

###### July 30, 2023

### A Universe of Numbers

**How Math Greatly Affects our Daily Life and Development of Civilization**

Math has worked WONDERS for our daily lives. You may not personally enjoy the subject yourself, and that’s okay! However, it’s difficult to disregard just how influential math has become in modern society, and the development of civilization as a whole.

It’s deeply ingrained in everything we’ve built, negotiated, engineered, and modernized. It provides order in our lives and minimizes the chaos. Our statistics, logic, ethics, and deeply encoded practices all have math to thank. And whether we see it or not, math materializes into phenomena all around us.

**Math is a globally understood and utilized knowledge**, which opens us up to connecting in ways we typically couldn’t. When we encounter an opportunity to have an exchange of knowledge, we can only grow in our comprehension. That’s probably why so many of our biggest advancements came into fruition when a great discovery in math was made.

I mean, take a look at the ancient Egyptians… once they discovered how to apply their mathematical findings, the Great Pyramids came to be and, well, the rest is history.

Even when we aren’t in active math mode or solving something directly in front of us, it’s a skill we implement daily. It forges our cognitive, intellectual, and academic discipline- and yes, we use it in domestic situations as well! What do you think lays the foundation for things like cooking, telling time, even filling our homes with furniture?

What are the benefits of math in daily life? Oh, let us count some (of the infinite) ways…

**Daily Benefits of Math**

**Builds Mental Discipline**: It’s no secret that math is an incredibly rigorous topic to learn and master, which we’ve been exposed to from very young ages. The numbers, equations, formulas, proofs, measurements, shapes… they are all intricately designed with their unique set of rules and methods. When we are presented with a problem to solve or decipher, it’s on us and our discipline to consistently show up and remain engaged. In order to get the answers we need to understand the math, that involves discipline and perseverance to get it done.

**Can Help Build a Growth Mindset**: This is especially useful for young minds & students, since they don’t have as much emotional or mental regulation. Our brains naturally react to a roadblock in two ways: find another solution or give up. When math presses us with a similar kind of obstacle, it gives us the option to learn the technique effectively by trying an improved strategy or settle for the wrong answer. Encouraging younger students or those starting out in math to find a new means of solving the problem helps stretch and grow their brain by LEARNING from prior mistakes. If we don’t learn, how can we be expected to do better?

**Logical + Quantitative Reasoning**: These are two fundamental elements to developing an argument. Providing the logical explanation or function combined with quantitative approaches can establish necessary evidence to prove the validity! Combining these both will cultivate inductive, deductive, even statistical methods. When applied to math, this can aid in analyzing the deeper meanings that the math insinuates.

**A Language of Science + Technology**: Although Science and Technology are their own innovative fields, math helps to organize and communicate their information. When math is applied to technology, it can help people such as scientists and engineers create systemic and productive practices to transmit useful knowledge. It also helps specify what is necessary to construct the tools and advancements making our daily lives that much easier.

**Foundation for MANY fields**: Aside from the obvious, math lays a strong infrastructure for many other fields. For instance, math is utilized in cooking, architecture, politics, economics, accounting, finance, mass production, and so many more fields. Jobs in ALL of these fields most likely require a background in math, being that they all use different disciplines within math to find what they’re looking for.

**Paves Way for Invention**: Obviously humanity would not be where it is without math to help them modify their inventions. With the use of math (and even a growth mindset), we can find new ways to strengthen or improve an idea or physical creation. If you read early Greek works revolving around the understanding of the Universe, they resorted to math to support their philosophical discoveries.

**Helps Us Understand The Universe**: Yes, math pervades all forms of universal matter. Even quantifiable amounts of sub particles can alter the impact of a singular atom. Things such as sacred geometry, quantum physics, and mathematical structure that explains all things discoverable (and still not discovered) in our Universe.

**Encourages Critical Thinking**: When it comes to math, critical thinking implies there is a reasonable decision or judgment needed to be made to execute the problem. In other words, the person doing the math must make thought-out decisions given the information and mathematical criteria at hand.

###### August 18, 2021

### Is zero a rational number?

If you are currently studying integers in your math class, you may find that rational numbers are the next topic addressed by your teacher. They may ask the class “is zero a rational number?” Although you know the definition of a rational number, this question can stop you in your tracks.

Knowing basic properties regarding integers and rational numbers may seem like a one-time lesson in your mathematics career, but being able to fully comprehend the purpose and the rules of rational numbers can directly help you with other complicated math concepts. Instead of just thinking the old narrative that learning a math lesson will only pertain to your specific lesson, quizzes, and tests, truly understanding rational numbers can help you learn how to solve multi-step problems in the future.

In addition, learning rational numbers and mastering integers helps you to understand how these math concepts all relate to one another as you progress in your education. Learning the relationship between rational numbers makes mathematics easier to understand as a whole.

By using critical thinking skills to find answers to questions such as “is zero a rational number?”, you can help simplify complicated equations and find the solution.

**First – what is a rational number?**

Before you can determine if zero is a rational number, we need to brush up on your definition of rational numbers. A **rational number** is any number that can be presented as ** a** over

**, where**

*b**b*is not equal to zero. Like this:

Therefore, practically every fraction that you have worked with in the past can be defined as a rational number. The only condition is that the “bottom” number in the fraction – also known as the denominator – cannot be equal to zero. Therefore, all whole numbers are rational, since any rational whole number can be expressed as a fraction with 1 in the denominator slot.

But how can you identify rational numbers from irrational numbers? Irrational numbers are not able to follow the rule that it can be presented as **a** over **b**, where *b* is not equal to zero.

In this case, you might come across irrational numbers that look like the following:

When looking at a fraction, sometimes the numerator can be zero. This means that **a** can equal zero, but *b* is not equal to zero. For new learners, just seeing a zero in the equation can sometimes throw off their thought process. However, keep in mind that a rational number with zero as a numerator can easily be simplified as such:

Since the greatest common denominator between 0 and 12 is 0, you can simplify a rational number fraction with a zero in the numerator by dividing both integers by zero. If you see a fraction with zero in the numerator, the answer will **ALWAYS** be zero.

**Examples of rational numbers **

Some examples of rational numbers include ¼, ⅔, 0/1, 8/6, and so on. As you can see, one of the numbers included here was 0 over another number – this is considered a rational number since the denominator is either a positive or negative number and is not equal to zero.

**Operations on rational numbers**

Just because a number is a fraction or contains a zero within the fraction, this does not mean arithmetic operations are impossible. Arithmetic operations are the basic processes of functions you can use with integers, including addition, subtraction, multiplication, and division.

- With addition, you can add two fractional rational numbers by making the denominator the same value.
- When carrying out subtraction, make the denominator the same number before subtracting one rational number from the other one.
- In the case of multiplying two rational numbers, make sure you multiply both the numerator and the denominator of the rational numbers.
- When it comes to dividing a fraction, you must calculate the inverse of the rational number. In this process, switch the numerator and denominator.
- For example, 2/4 will then switch to 4/2. You can double-check your work by multiplying the two values together to see if they equal 1, with the equation being (2/4)x(4/2)=1.

**What are irrational numbers?**

There are clear differences between rational and irrational numbers. Now that you know rational numbers are any type of integer that can be expressed as a/b where b does not equal zero, you can narrow down all integers that do not follow this formula to be considered irrational numbers.

Rational numbers are either positive numbers, negative numbers, or equivalent to zero. However, irrational numbers cannot be written in the form of a/b, but must be written as a decimal.

One of the most common irrational numbers is Pi (), which has endless digits after the decimal point. If you have studied geometry in the past, you know that Pi is the ratio of the circumference of a circle to its diameter – a number that comes out to approximately 3.14. However, this solution is just an approximation, but not accurate enough to be written as a fraction or classified as a rational number. Since the decimal digits of Pi are nearly infinite, you **cannot** use a simple fraction that represents Pi.

Furthermore, Euler’s number () is another irrational number that cannot be simplified or written as a fraction. This number is the base of natural logarithms and is an important concept used in the exponential function. Euler’s number is typically written as 2.718, but contains over 1 trillion digits in the decimal – making it impossible to compose in fractional form.

Another common irrational number is the Golden Ratio (), a ratio called “divine proportion”. This unique ratio is a common number that is found in various aspects of our life and culture, not just in mathematics. Appearing in paintings, artwork, sculptures, and buildings, the Golden Ratio is known for being aesthetically pleasing. This number is typically approximated as 1.618, but cannot be expressed in simple fraction form.

**So – is zero a rational number?**

Now that you know the difference between rational and irrational numbers, you may still be asking yourself – is zero a rational number? You know that zero cannot be the denominator of a rational number, since this would classify the rational function as undefined. However, zero can be the numerator in a fraction, which qualifies zero as an eligible rational number.

Since zero is an integer, it can be written as A in the formula A/B (ex: 0/1, 0/3, 0/6, etc.). Writing zero as the numerator over a positive or negative denominator confirms that zero can be considered a rational number (ex: 0/2, 0/-2, etc.).

Keep in mind that 0 cannot be used as a denominator in a rational number equation. If this was the case, any number divided by 0 would result in infinity – as we learned earlier, irrational numbers cannot be written as a fraction with integers as both the numerator and denominator. Any integer divided by 0 is an irrational number.

**Why is it important to know if zero is a rational number?**

If you are struggling in your math class and you have **finally cracked the code** to the rational vs. irrational number debate you might still be wondering how this will help you understand difficult math concepts.

Knowing that zero is a rational number can give you deeper insight as to how integers can be calculated and how they function within different equations. Being able to distinguish between rational and irrational numbers can help you understand more difficult concepts that you previously had a difficult time understanding.

Learning rational numbers, understanding how to interpret rational numbers, and identifying rational vs. irrational numbers can help students understand how to represent rational numbers when composing equations, how to determine which fractions are greater than others, and how to simplify rational numbers in fractional form.

###### July 23, 2021

### How To Find A Statistics Tutor

“As with almost anything, you benefit most by being taught by someone who has a solid knowledge of the fundamentals, has real world experience in the area, and has the ability to communicate effectively.”

― Ron Glaser, P.H.D., retired US Government statistician and retired UC Davis Statistics Professor, on finding the right statistics tutor.

If you are looking for a statistics tutor, there is a good probability you are looking at a math syllabus full of unusual words: biostatistics, linear models, regression analysis, data mining, survey sampling—that kind of thing.

Statistics isn’t like algebra or geometry, it’s a whole other animal. Even calculus professors can be lousy when it comes to statistics. And if you need help, you may find your math tutor doesn’t make the best statistics teacher either. So, where do you find the elusive statistics pro? Craigslist? The local college career office? Is there a young, fun descendant of the Father of Modern Statistics Sir Ronald Aylmer Fisher living in Portland?

If you actively seek out a statistics specialist, there’s a ninety-nine percent chance you will find what you’re looking for. But if you want a 100% guarantee, here’s what you need to do…

**Where to Find a Statistics Tutor**

If you sense you need help in statistics, you are already ahead of the curve. Many struggling students can be too stubborn to get help and end up tanking their grade point average by attempting to take on statistics on their own. If they pursue a career that uses statistics, such as sales or computer programming, they might be at a loss, frantically searching the internet for a review course. But it doesn’t have to be that way!

It’s important for anyone seeking a career that involves statistics to find a statistics tutor with a well-rounded understanding of both statistical computations and how it relates to a career in data science or statistics. The best place to find private statistics tutors is online. Don’t be overwhelmed with all the tutoring websites out there because there are ways to trim the fat and find the perfect fit.

Look for websites that have been around for a while, have helpful contact information, and use official email addresses or phone lines (as opposed to tutors’ personal contact information, which can be a sign of inexperience as a business). Another trait of a trustworthy tutoring site is a review section where users can provide feedback on their tutoring experience. This way you can validate a good reputation within your local tutoring community.

**What Qualifications Should Your Tutor Have?**

According to Ron Glaser, a retired Lawrence Livermore Lab statistician and university statistics professor, it’s ideal to find a tutor who has a degree in math or statistics, or a college student with marked success in statistics coursework. Glaser would not recommend engineers or scientists who have not had formal statistics training as tutors, because they tend to bluff on knowing lessons students need, but don’t actually know themselves.

You will need someone who can prepare you for college-level coursework in statistics, which translates to someone who has at least a B.S. in statistics or a closely-related field, such as biostatistics, applied mathematics, or computer science. If you are searching on the Internet, think *grad student*. If a tutor is actively working toward a Masters or PhD in statistics, they will have the necessary experience to teach course material and apply it to everyday life.

However, having academic experience isn’t always enough. Consider finding someone who has experience applying statistics in the workforce or has experience teaching or tutoring. The combination of education and experience in the field will be your best bet for a statistics tutor.

To take the search one step further, here is a checklist for just some of the key elements we look for when hiring statistics tutors at Tutor Portland:

- Can this tutor explain tough concepts in five different ways?
- Can this tutor use metaphors that relate to the student’s life?
- Can this tutor adopt an active approach with in-depth discussions about statistics?
- Does this tutor embody integrity and virtue?
- Can this tutor effectively teach us, the Tutor Portland team, before we hire them?

**How much does a good statistics tutor cost and how to ask?**

Okay, so you found yourself a qualified tutor named Sir Ronald Aylmer Fisher Jr. and he’s fabulous, but you need to know how much he charges per hour. Here’s a statistic for you: The cost of living in Portland is 29 percent higher than the national average. On a college student’s salary of $21,000 per year, with a meal card from the folks, can you afford a tutor?

**Here’s the rundown on the costs: **

At Tutor Portland, for instance, we offer online and in-home tutoring backed by a Better Grades Guarantee for our Portland peeps with different budgets. Let’s say you need a statistics tutor for your coursework at Portland State University. We charge $96 per hour for our Silver Plan academic coaching or $384 monthly for one hour per week. Your hours never expire, rolling over to future months, and the more hours you book, the more discounts you receive.

If you decide to hire an independent tutor without such payment models, you should consider asking their rates upfront. The more education, work experience, and overall skills the tutor has, the higher the rates may be. And don’t be shy— after all, statistics tutors are used to talking numbers.

**How to Get the Most out of a Tutoring Session**

It’s all about your personal needs. So the key to maximize your time with a tutor is to study between sessions. By being observant (and hopefully excited) about how statistics play out in the news, in school, in work-life, and at home, you will be mastering the subject in a meaningful way. Then when you meet with your tutor to review the lessons, you can have a deeper understanding of the course material in less time.

**Now that You’ve Aced Statistics … **

If statistics clicks with your brain after a tutoring course, you may even choose it as a career. Among the industries that hire statisticians, the median wages range from $70,000 to more than $100,000 per year. Statisticians are in high demand, and according to Northeastern University, employment for mathematicians and statisticians is expected to grow 30 percent from 2018 to 2028.

Now that you found your tutor who is qualified, affordable, and can get you ahead in your statistics class, what’s next? Maybe another class! TutorPortland offers tutoring in test prep, science, Spanish, and English, so contact us if you’re looking for some extra help.

###### March 24, 2021

### How Tutoring Can Help Students Master Math in the Era of Virtual Learning

If your kids are taking math online right now, I can see why you might be worried. It’s hard enough to hope for As in math classes when kids are attending school in person, but 2020 made everything more complicated.

Now that class is online, it’s even easier for kids to find themselves falling behind. Students are more distracted and less engaged. Class discussions are virtually nonexistent. Getting individual attention from a teacher seems harder than ever. How in the world are students supposed to master complicated math concepts in this environment?

While the online format isn’t ideal, tutoring can help! For kids who are missing that one-on-one interaction, a math tutor can bring them the connection they crave, now from the comfort of their home. Classes are now almost always recorded, giving tutors a way to glimpse course material and work through it with students. Virtual pod-based tutoring can help kids supplement the social aspect they might be yearning for. So if you worry about your kid grasping the math curriculum, don’t give up just yet. Here’s how tutors can make a difference…

**Engaging Students**

Many parents worry that Zoom is less engaging than in person math class. There’s definitely some truth to that–it’s hard for kids to get invested in the material when they’re just another black square in a sea full of turned-off cameras. It’s tempting for kids to start playing Minecraft or scroll through Instagram during a lecture as there’s no teacher in the room. Discussions aren’t as fluid or active and it’s hard for kids to make themselves known. In this virtual environment, students are not as involved in the subject at hand.

However, getting a great tutor can help solve this problem. No longer will your kid be an anonymous member of a huge Zoom class. Tutoring gives kids a personal, individualized approach to learning, even through the internet.

At Tutor Portland, we give one-on-one tutoring that’s totally focused on engaging the student. Our tutors build a strong rapport with students, ensuring that kids feel safe to express their frustrations and explain their own way of thinking about problems. Tutors check back with students each week and give them the positive encouragement they need to strive for growth. Even though kids have to learn through a screen, with this approach, we can keep kids excited and interested in the material.

That being said, even if students remain involved, they might not be able to be able to keep up. Math is cumulative, meaning if students miss one concept, they can easily get behind. Working through recorded lectures with a tutor might be able to change that…

**Keeping Students From Falling Behind**

Fast-paced math lectures are so common, it’s a wonder how kids can keep up at all. This has been a problem long before online learning. It’s a byproduct of our tendency to teach math with little to no discussion among students, let alone individual questions and feedback. Now that many lectures are virtual, students might have even less opportunity to slow down the class and inquire about a tricky concept.

Luckily, in this new online educational environment, lectures are almost always recorded. These recorded lectures are made available to students at any time. This means students can go back and rewatch whenever they need to in case they get stuck on homework. They can pause and rewind or go to a specific section when they may have overlooked a detail. Plus, if they want to review for a midterm or a final, they have every lecture laid out for them online. Instead of having one chance to get the teacher’s lecture, they can access it any time.

This is a great tool that tutors and kids can use together. Previously, if a kid struggled to comprehend an idea, tutors would have to guess how the teacher taught it to them. Now, tutors can see exactly how the concept was explained to the student, so the tutor can more deeply understand their student’s learning process. Students and tutors can go through recorded videos and take a new look at any material the student doesn’t quite understand.

Beyond having relationships with teachers and tutors, students also need to relate to each other and work together. But, how can tutoring help kids get that experience without in-person classes?

**Incorporating the Social Aspect**

Although socializing with others may seem extraneous to learning equations, having discussions with others about material really helps struggling math students. Studies show that students understand concepts better when they’re able to verbalize their thinking and have back-and-forth exchanges with other individuals.

However, online school makes it impossible for kids to talk about material as they walk the halls, and makes forming study groups a lot more challenging. This is especially true if your student is new to the class or doesn’t have a group of classmates to collaborate with.

Tutoring can help by not only connecting students with a great tutor, but a network of their peers as well. My brother and I had this social component in mind when we created **Zoom Tutor**, a website made specifically to help kids through the online educational landscape. On Zoom Tutor, not only is one-on-one tutoring offered, but pod-based tutoring with a group of students as well.

This means students are organized into small groups that meet each week to go over material. Although they may not be able to gather in person (yet), these regular sessions give kids the ability to meet and work with an entirely new group of peers. Instead of working through problem after problem alone, they can have a community to share their frustrations and victories. This can fundamentally transform students’ attitudes towards math. Instead of stale repetition, these peer groups help make math fun and social.

**Get Your Kid Tutoring Today!**

Even though learning math online may seem like an impossible challenge, tutoring can help students make the most of it. When Zoom leaves kids uninvested, good tutoring can help them engage with material again. Working through recorded lectures with tutors can guide kids towards success if they find themselves falling behind. Incorporating group work into the online tutoring format can help students have meaningful discussions that allow them to get a grasp on complicated math subjects..

So if you want to help your kid make the most of learning math online, you can sign up for your first free Tutor Portland session **here**, free of charge. You can also check out Zoom Tutor **here**! If we can give students access to a meaningful learning experience despite the tribulations of the online format, we can help kids come out of this challenging time on top, ready to achieve their goals.

###### December 15, 2020

### The Ultimate Guide to Math Placement Tests

So your child has completed their college apps and they’ve started to narrow down their higher education choices. Whether they want to attend a junior college, attend school part-time, or are taking the plunge into a four-year university, their big decision will help shape their future. You’ll want to be sure that they’re as prepared as they can be.

Before they even step foot in a classroom, their skills will need to be tested. Math placement tests are a required examination prior to enrolling in specific classes. If they’re worried about what lies ahead for them when it comes to math placement tests and scheduling math classes, there’s no need to worry.

We have the ultimate guide right here for parents and students like you to better understand what math placement tests are all about.

**What is a Math Placement Test?**

A math placement test is designed to measure a student’s math skills and gauge the most appropriate math classes they should take for the upcoming semester. Before starting college or university, students must complete a math placement test at home. This happens after a student has been admitted to a school and is a normal part of the enrollment process.

There is no passing or failing a math placement test. The point of these exams is to see how competent the student is in the subject. They are more of an assessment of personal skills rather than an analysis of mastery. A math placement test is not a measure of intelligence, but a measurement of personal experience and how well a student demonstrates that experience.

After the test, the school will tailor a choice of math class to the student’s best strengths. If a student is a top scorer, they’ll be rightly placed in advanced classes that will properly challenge them. If they score lower on the math placement test, then they will be placed in less intense math courses.

**What to Expect on a Math Placement Test**

Although all schools require students to take a math placement test, there is no universal standard they adhere to. Each university or college will create their own tests that best measure math skills according to their own set of standards.

However, there will be some similarities across the board. Questions will be pulled from a wide variety of math topics such as algebra, geometry, trigonometry, and some precalculus and calculus, but the questions will not dive deeply into each subject. The range of questions will be wide, not deep.

The wide-range of questions about basic concepts is followed by longer word problems. These word problems focus on the application of concepts. Students will apply the previously mentioned concepts to help solve longer story problems. Further, the test will ask students to provide written analysis to test if they can fully understand and demonstrate mathematical concepts.

These tests usually do not have a time limit and are mostly multiple choice. The point of the exam is to measure skill, not speed. They’re usually administered online too, so there is no need to travel to a testing location. The tests are often completed from the comfort of a student’s own home.

**What’s The Deal with High Placement Scores?**

Preparing for math placement tests can save students and parents time and money down the road. If your university requires payment for classes by the unit, students can save money by testing out of classes that would otherwise be required for their degree.

By scoring well, students can bypass entry-level courses and qualify for more challenging (and more interesting!) math classes. By scoring poorly, the school may place students in lower intensity remedial courses and take time away from more enriching classes.

If a student plans to enter a field that is less math-intensive, they can bypass math classes altogether if they score high enough. By testing out of basic math classes, they free their schedule to take other classes that are more relevant to their field of study. This will save time and money as students will skip a few steps on their way to complete their degree.

**How to Prepare for a Math Placement Test**

Before taking a placement test, I encourage students to brush up on basic mathematical concepts. A math placement test will have questions about basic arithmetic such as addition, subtraction, multiplication, division, fractions, proportions, averages, decimals, and integers. It sounds like a lot, but it’s nothing students haven’t mastered before!

Here are our tips to **maximize your studying habits!**

However, it’s important not to put too much weight on the outcome of the test. There is a downside of over-preparing. If students prepare for the test in anticipation to score “well” or “ace” the exam, then they could be setting themselves up for a difficult situation. By studying for hours to ace the exam, students may earn placement in an advanced level math course they are not ready for. They might become overwhelmed by the higher level material because of their inflated placement score.

Remember, there is no passing or failing these placement tests, so over-preparing won’t always pay off. Ending up in a less intense math course might be the perfect situation for a student who isn’t as mathematically gifted as their peers.

Conversely, if a student enters the math placement test completely unprepared, then they could be placed in a math course that is far below their skill set. These students might get stuck paying for a class that is repetitive, boring, and ultimately a waste of their time because they did not demonstrate higher level math skills on the placement test.

**Final Thoughts on Math Placement Tests **

If a higher education institution accepts you or your child into their program and requires you to take math placement tests, look into hiring a tutor. When you hire a tutor to work one-on-one, you can focus on reviewing the areas that are the best use of your time. Meeting with a tutor to hone math skills could make a positive difference for your future in higher education.

Experts from **Tutor Portland** or **Zoom Tutor** can assist in tailoring a plan for you or your child. Having a tutor that understands how the math placement test works will make a huge difference in your educational experience. By fortifying math experience, you will be ready to take a math placement test and best serve your educational career.