Any ideas on how to teach fractions?

Perhaps this wold help. I didn’t watch the whole thing but I don’t think there is a simpler way to explain it. https://www.youtube.com/watch?v=-6AhN38OR14

I just skimmed the link. One thing they did that I suggested was to find ways of comparing the size of two fractions in simple cases.

If you can, use lots of manipulatives. They are one of the best ways for students to begin to grasp fractions.

I don’t think there’s any point or joy in teaching or learning fractions before deeply understanding what they mean, and in my experience with adults who don’t get fractions, that’s missing.

So I’d do physical examples, then add what the language and notation for them is, staying with the smallest numbers (i.e., half, third, fourth, ½, ⅓, ⅔. ½ + ½ = 2/2 = 1. ⅓×3 = 3/3 = 1. ¼ etc.) and then slowly working up, advancing only when the relationship between the physical thing and the numbers/language is fully understood and automatic.

If they cook, refresh yourself on the measuring system they use (conversion between tsp, tbsp, fl oz, cups, pints, quarts, gallons) and do that after the basic fractions, since they may already have a working relationship to those physical measures.

Otherwise you’re just teaching a weird abstract number game governed by secret logic that doesn’t make sense to them yet.

@Zaku , You make a good point. One thing I think is an obstacle is the understanding that ¾ can be understood in two equivalent ways.

The usual way of representing ¾ is to take a rectangle, cut it into four equal pieces and then take 3 of them.

I could also take 3 identical rectangular pieces, make a large rectangle out of them and then divide the large rectangle into fourths. Each of these fourths is ¾ of the original rectangular pieces. This second interpretation is important to understand. If you think of ¾ as a division problem then we are dividing 3 by 4.

Intuitively it is not obvious to me that the two interpretations are equivalent. I might be able to write a formal proof that this is the case, but that is not going to do any good for my student.

Yep. Try doing it with physical objects they can hold, and ask them to hold them as they do it.

Your example can be done with four squares or blocks:

[_][X]
[X][X] equals [_][X][X][X]

If they have blocks to hold, it can start to make sense in an entirely different way, though physical intelligence. Adults often know things about fractions but their mathematical language doesn’t map (with their current thinking about it) to what they know.

One could also play with water or rice or something, unless that doesn’t appeal.

Or move on to pictures. There are some pretty good visualization available by Googling things like “fraction visualizations”, and some confusing ones or overly complex and/or child-patronizing ones.

To turn what you were just suggesting into physical exercises, I think that this one and this one, especially if they were printed and cut out and held, could be used to easily give a physical understanding of what numerator (the number of pieces) and denominator (the size of the pieces) are, as well as what adding, multiplying and fractions are physically and on paper.

Money and cooking are the best ways to explain the very basic concepts regarding fractions.

A 25¢ piece is actually called a quarter. Why? Because it is a quarter, or ¼, of a dollar. Why? Because you need 4 quarters to make a dollar. The 4 is the denominator. How many quarters you actually have is the numerator.

A dollar is 100 cents, or 100 percent. A quarter and a quarter equal half a dollar, or stated as ¼ + ¼ = 2/4 or ½. If you forget the rules you can rethink them through logically if you think about things like money. Get both denominators the same and that stays constant even on the other side of the equal sign, then just add the numerators.

As far as multiplication and adding fractions in general, I tell people all the time when it comes to math, stop worrying about understanding it, just learn the equation. Memorize the rules. This doesn’t work for very complicated “word” problems, but for basics like adding, dividing, multiplying, basic algebra, it does. Later the understanding falls into place. The thing I hate the most about the new way math is being taught is adding all the reading and word problems they are doing know at very basic levels.

The advantage you have with your student is she is an adult, and has real world experience using math and she probably doesn’t even realize it.

Back to money. 2/4 (two quarters) times 3 equals 6/4 (six quarters). Literally, have coins with you to think it through. It will demonstrate how to write out the equations and the rules for multiplication.

Using pies and cakes is good to. A pie cut up into 12 pieces would make the denominator 12. Cut into 6 pieces the denominator is 6. How do you write both pies as a fraction if you want to represent half a pie?

What if the pie is 5 pieces or 6 pieces. Which is smaller 1/5 or 1/6? Draw it out.

Another thing that really helped me was turning fractions into a decimal form. That’s how I was able to check myself sometimes when I was first learning fractions, but that might be complicated for some people?

As far as cooking, ¾ or a cup of sugar and ¾ cup brown sugar is how much total sugar? 6/4.

@JLeslie , I appreciate what you are saying but as a lover of math, it pains me to say, just follow the rules. Whenever possible, I try to give a plausible explanation. For example, to explain the inversion rule for fraction division, I represented a/b divided by c/d by making a fraction with a/b in the numerator and c/d in the denominator. To clear out the c/d term, multiply both the numerator and the denominator of this strange fraction by d/c.

^^I am a lover of math and excelled in it in school. people who are terrible at math always seem worried to me about understanding it.

Just a few years ago a friend of mine was teaching 7th grade math. She had been an IT corporate person for years, and for the first time she was a teacher. Anyway, I’m talking to her husband and he quizzes me and says, “JL what’s seven minus a negative 2.” I tell him 9. He says to me, “how do you know that?” How can a negative be a positive? You know what? It doesn’t matter, just memorize two negatives are a positive in subtraction and you’ll get the answer right. Instead he worries about why and can’t answer the question, and sucked at math his whole life.

Once in a rut that you feel you can’t do math, then you get the mental block about it. When I took business calculus I needed to be able to set up the problem correctly, but doing the algebra is just knowing the order of operations and the rules for algebra. ¼+1/6 is basically knowing the rules. Knowing what ¼ and 1/6 look like is the understanding, but not necessary to solve the problem. You’re a math person and easily picture those measures and move numbers around, me too, but you are dealing with someone who doesn’t is my assumption.

In my answer above I gave an example of using money, cooking, and the old fashioned pies for understanding, so I didn’t completely leave out understanding.

When I was a very little girl my grandpa drew pies in the sand one day on a beach in Cape Cod to teach me fractions. I didn’t completely “understand” it. A couple of years later we were taught fractions in school, and what my grandpa had shown me fell into place. Math is often like that. You struggle with chapter one, but then by the time you’re on chapter three, chapter one is easy and makes sense.

I have an idea for how to teach addition and subtraction of positive and negative numbers. Many people do not realize that if we limit ourselves to addition and subtraction, the rules do not distinguish between positive and negative numbers.

Imagine a balance scale and assign one side as positive and the other side as negative. To add two numbers, place a weight on the corresponding side of the scale. To subtract a number, place it on the opposite side of the scale. For the example of 7 – (-2), the rules I gave place the 7 and 2 weights on the positive side. If instead we had -7 – (+2), both the 7 and 2 would go on the negative side.