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Mathematical Journeys: The Unperformed Operation

Come with me on a journey of division. I have here a bag of M&Ms, which you and I and two of your friends want to share equally. I'm going to pour the bag out on the table and split it into four equal piles. For this example, “one bag” is our whole, and the best number to represent that whole would be the number of M&Ms in the bag. Let's say there were 32. If I split those 32 M&Ms into four equal piles and asked you how many were in one pile, you could certainly just count them. But a quicker way would be to take that 32 and divide it by the number of piles I'd made, which in this case is 4. You'd probably write that as: 32 ÷ 4 = 8 So there are 8 candies in each pile. Seems easy enough with a large number of M&Ms, right? But what if there were less candies – what if our “whole” was less than the entire bag? Well, for a while we'd be okay – if there were 16, for example, we'd do the same thing and come up with piles of 4 instead... read more

Math + Music = Elementary Fraction Success

By far, one of the most difficult concepts in elementary mathematics is fractions...and it is all our fault. One of the major misconceptions among many education systems was that early exposure to fractions would help students learn them. This meant attempting to introduce fractions before students could even multiply or divide. You have no idea the trauma this has had among decades of students. Education systems created self-induced math anxiety.   For years I had to address what I can only describe as fraction PTSD. I had talented Algebra students immediately clam up if the problem had a fraction. Now as a teacher I of course did my job and we spent time trying to get ourselves comfortable with fractions but in the back of my mind I knew I was using valuable class time to address an issue that simply shouldn't even rear it's ugly head in Algebra. But every year it was there. Students were crying, parents were crying, and teachers were crying over the fraction... read more

Math + Music = Elementary Fraction Success

By far, one of the most difficult concepts in elementary mathematics is fractions...and it is all our fault. One of the major misconceptions among many education systems was that early exposure to fractions would help students learn them. This meant attempting to introduce fractions before students could even multiply or divide. You have no idea the trauma this has had among decades of students. Education systems created self-induced math anxiety.   For years I had to address what I can only describe as fraction PTSD. I had talented Algebra students immediately clam up if the problem had a fraction. Now as a teacher I of course did my job and we spent time trying to get ourselves comfortable with fractions but in the back of my mind I knew I was using valuable class time to address an issue that simply shouldn't even rear it's ugly head in Algebra. But every year it was there. Students were crying, parents were crying, and teachers were crying over the fraction... read more

Better Ways to Add and Subtract Fractions!

In elementary school we are taught to add/subtract fractions in a way that, quite frankly, is a BAD WAY of adding/subtracting fractions! If you don't know what I'm talking about, here's a short review...   We have a fraction, say (1/2). Let's add a third. We have (1/2) + (1/3).    So, what's the first step?    Well, in elementary school you were probably taught to cross multiply. Let's try it   We first get (1)(3) = 3 and (1)(2) = 2 Adding these together gets us 3 + 2 = 5 Now we multiply the denominators, getting (2)(3) = 6 We can now put these two numbers together: (5/6)   Though this method works, it's not the best way to go about adding two different fractions.  First off, why the heck does this work?!   Though it seems like magic, there's a method.    First, we cross multiplied. Putting this in a way that shows the whole expression gives   (3)(1) + (2)(1)... read more

Fractions, Decimals and Percentages

When working with fractions, I find it effective to require students to convert each fraction that we work with to its decimal equivalent, to convert that decimal equivalent back into the original fraction, to convert that decimal into its percentage equivalent, to work a simple percentage problem using that percentage and finally to work the same problem using the initial fraction.   This comprehensive method helps students to see the relationships between fractions, decimals and percentages in a holistic way and to promote the necessary skills in each element.  

Mathematical Journeys: What's a Radian?

Buckle up readers, it's Trig time! Trigonometry can be scary to many students, and in my opinion, a lot of that is because one of the most confusing concepts in trigonometry occurs right at the very beginning, in the form of the Unit Circle and Radians. Let's start at the beginning. Give yourself a circle with a radius of 1.  Now center that circle on the origin of a coordinate plane, so that the line of the circle itself passes through the points (1,0) (0,1) (-1,0) and (0, -1). Got that? Now, this circle is referred to as the Unit Circle, because the radius is one unit and it is therefore easier for us to do various manipulations and calculations with it. Now choose any point on the circle (we'll call the coordinates of that point (x,y)), draw the radius to it (which will still be a length of 1), and drop a line back perpendicular to one of the axes. Do that and you'll have a right triangle with the... read more

Is there a way to change a fraction without affecting its value?

Yes, there is only one way.  Let's say for example that we have a fraction of 2/3.  Now, the bottom number is the denominator which means the number of equal parts into which a whole circle most specifically is divided.  So the circle is divided into 3 equal parts.  On the other hand, the top number is the numerator which means how many equal parts out of all of them are lightly shaded inside the circle. So 2 out of all 3 equal parts of the circle are lightly shaded.   Now, the only way to change the number of equal parts without affecting the fraction value is to multiply it by any number you want which will also change the numerator.  So let's say for example that in the fraction of 2/3, if you wanted to divide each of those 3 equal parts into 2 further equal parts, you will have a new number of equal parts which is 6 (3*2=6).  This will affect the numerator 2 as well since this is included in the total number of equal parts, so each of... read more

I never really learned that...

One of the common challenges for many Algebra students is forgetting important concepts from Pre-Algebra. So many students complain that they never fully learned fractions, decimals and percentages or ratios, rates and applying math to word problems. Without solid memorization of multiplication & division tables, factoring and simplifying are much more difficult. A better understanding of the basics, including learning different methods and shortcuts, can not only boost confidence but can improve grades and SAT scores. Spend time with a tutor or use different websites to review topics from previous years. It will help, exponentially!

Teaching Children Fractions

Mothers generally know this trick. It works especially well with food children do not want to eat, but must. Tell the child that he only has to eat 3 bites. Then let the child eat just that many bites, which you can count together if you like. Increase the number of bites as the child learns to count. Alter the exercise with how many peas can be left on the plate; how many bites can be exchanged for another food or desert, and other tricks. As the child learns fractions, work with eating (or leaving on the plate) 1/2 of the food, or 3/4, or other familiar fractions. Fill glasses 1/2 full of their favorite beverage and offer another 1/3 or 3/4 or so more when he drinks the first fractional amount. Conversations and expectations and games like this applied to food and drink, picking up toys, helping out around the house, etc., help children from ages 5-7 develop their number sense. These tricks can be used just about every day for a few minutes a day--longer only if... read more

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