News & Updates
August 18, 2021
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 b, where 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.
April 23, 2021
When you’re trying to memorize hundreds of conjugations and spend hours working to get the pronunciation just right, you may feel like mastering Spanish is impossible. Writing and reading in a foreign language takes a ton of practice, and speaking the language aloud fluently is even harder. It requires students to conjugate and translate faster. In a big classroom full of people, it’s easy for students to get lost in a sea of faces or intimidated by their peers. They may not get the individual attention they need to truly grasp the language.
But the value of understanding Spanish should not be underestimated. The benefits of mastering this commonly spoken language extend far beyond the academic realm. Learning Spanish allows us to connect with Spanish speaking friends and neighbors. It empowers us to travel to different parts of the world where Spanish is the primary language. Gaining the skill of speaking a second language is also great for the brain, increasing memory and ability to focus.
So how can we help kids truly master Spanish when the classroom just isn’t cutting it? Tutoring is the saving grace you’re looking for. With a tutor, kids can have a much more individualized experience. Students can have fluid conversations with tutors in Spanish to get the back and forth speaking practice they need. If they feel self conscious in front of the class, they can get comfortable talking in Spanish with a trusted tutor. Tutoring can also switch up boring routines for students, to get them out of the mundane workbook mentality.
If students are quite grasping the material, here’s the top three ways tutoring will help them Spanish like a pro.
Cementing Speaking Skills
Tutors allow students to drive home their speaking skills by having fluid back and forth conversations in Spanish. Learning a language is not like picking up math, science, or even English. It requires more than just rote practice. Students need to have frequent conversations in Spanish to really be able to translate with speed.
But learning in a classroom with thirty other people makes it hard for students to get those conversational skills. There’s not a lot of chances to have one-on-one conversations with their teachers, as there are so many students and only so much time.
Tutoring brings a personal component that’s missing in the classroom. Students can have discussions in Spanish with a tutor who is fluent, even doing whole sessions in Spanish once kids get to a certain level of ability. A tutor can point out exactly where kids are missing the mark in these conversations and help them really focus on changing key mistakes. Plus, tutors can educate kids about informal nuances in the language that a textbook might not mention.
I know our Spanish tutors are effective because I have every Tutor Portland tutor teach me before I hire them. With this process I’m able to see which Spanish tutors are able to break free from basic repetition exercises and have more sophisticated, productive conversation. Not only that, but Tutor Portland emphasizes principles like patience and conceptual learning, which promise to make the most out of Spanish tutoring sessions.
Self Conscious Students
Speaking Spanish regularly with a tutor can help kids ditch performance anxiety when it’s time to translate in front of the class. Being fourteen years old and having to speak to a class full of your peers is hard enough. Now try doing it in a language you barely know! One of the biggest barriers to students becoming fluent Spanish speakers is their anxiety about speaking in front of a class.
We want to ease that anxiety. One of our most important missions at Tutor Portland is to create a deeper bond between student and tutor. Our tutors go beyond just a transactional relationship with their clients, but instead act like mentors. This emotional bond helps establish trust. This trust will be a key component for students attempting to learn Spanish, as it will help them overcome anxiety about speaking.
With a tutor, students can practice with somebody they trust. They can work out the kinks with the same person every week, without being embarrassed. With some positive reinforcement from a tutor, they’ll gain confidence. They’ll be able to head back into the class ready to run through conjugations like a pro in front of everyone.
Escaping The Endless Flashcards
Spanish tutoring creates a more engaging curriculum than boring flashcards or workbooks. While these tools help students memorize words and phrases, they don’t really allow teens to master Spanish. They’re mostly used for teaching the subject matter to large groups of students when the teacher can’t get to each person individually. Unfortunately, they don’t give students the mindset to genuinely understand the language.
Luckily, one-on-one tutoring helps with this too. Tutors can give students new and interesting ways to encounter the language, like games or unique activities. They can assign students fun writing prompts based on the student’s own interests to get kids excited about learning. Giving frequent positive reinforcement helps kids feel personally fulfilled as their Spanish language skills grow.
At Tutor Portland, adopt a growth mindset and want to help students always be improving and changing. Beyond just boring and unhelpful activities, we create a personal connection with students and use that bond to give them truly effective teaching, so they can strive towards improvement.
The Perfect Tutor is One Click Away
If you want kids to master Spanish, tutoring might be the way to go! A tutor can give your kid the practice they need to speak fluently and confidently, and allow them to break out of a boring class curriculum. With Tutor Portland, you can be certain that your child is getting the personal, effective tutoring that they deserve. Sign up for your first session, free of charge.
July 1, 2018
Research shows that, over the summer, Portland high school students are likely to experience summer learning loss. Students, on average, lose the equivalent of 2.5 months worth of math education during the summer months. Decades of research (dating back over 100 years) prove this stunning fact.
However, some students actually report slight gains in learning and brain development over the summer. Why? They read books. Work with tutors. And attend educational summer camps. (My mom used to send me to these. Thanks, mom!)
Tired of summer learning loss? You better be! Research shows that it’s cumulative. That means each summer piles on-top of each other, weighing down your child’s grades like a heavy rock.
Quick Tips on helping your Portland high school student prevent summer learning loss:
- Go To The Library
- Reward Them For Khan Academy
- Teach Them About Finance and Investing
- Read To Them
- Solve Math Problems With Them
You could consider creating a reward system, whereby if your child reads for 30 minutes (or studies math using Khan Academy for 30 minutes) they are free to play with their friends for the afternoon.
Remember, this is also important for standardized tests. Research shows that after the summer months, students score lower on standardized tests (than they did at the end of spring). And this loss is cumulative. For example, in math, students who don’t learn in the summer can end up 2.5 to 3 years behind their peers. Could this be a reason that your teenager is struggling in algebra, geometry, or calculus? In our experience, we see this happen all the time.
Is your child on the bubble?
Summer learning loss is especially important if your child is struggling or “on the bubble” in any of her classes. A few hours in the summer can mean the difference between a D or a B in the fall.
Ask your teacher for help. Often, teachers will have extra resources and ideas for how their students can catch up and improve in the summer. Ask about: workbooks, online resources, or fun activities.
Checking with next year’s teacher is often an even better idea because they can help give your child a direct head start.
Other ideas to consider:
- Summer Writing Courses
- Dance Classes
- STEM Summer Camp
Planning a lot of vacations this summer?
Try to find some teachable moments. Let your child help plan and research the trip. Take them to the zoo, library, or a museum. Encourage them to write a journal during the summer.
I know a number of parents who suspend allowance during the summer… and instead reward good behavior (studying, reading, learning) with prizes or compensation. Research shows that these rewards don’t have to be big to work. They just have to be timely.
Good luck on your summer learning and fun!