### Equivalent fractions - a visual model of splitting the pieces further

With learning fractions, there is always the problem of "so many rules to remember". I offer this visual method of splitting the pieces further, and using the arrow notation as a remedy; hopefully this would help fix the method in students' minds.

Making equivalent fractions is like splitting all the pieces further into a certain number of new pieces. For example, if I split all the pieces in 3/5 into three new pieces, there will be 9 pieces. And, instead of 5th parts, they will be 15th parts. If you have an image and you split even the "white pieces" into three new ones, you'll see those 15 parts. So, 3/5 = 9/15.

The arrow notation shown in the video has one arrow between the numerators and another between the denominators. It also has a little "x3" written next to it. This is to signify into how many pieces we split the existing pieces.

This notation can help students not confuse equivalent fractions with fraction multiplication. The two fractions are equivalent or the "same amount of pizza"; one is not three times the other.

Please also see this free sample worksheet: Equivalent Fractions worksheet. This worksheet shows the same notation and the same idea as the video. It is a sample from my book Math Mammoth Fractions 1.

Please let me know what you think of this notation. I haven't see it anywhere else, but maybe it does exist somewhere. Do you think it confuses or helps students?

Making equivalent fractions is like splitting all the pieces further into a certain number of new pieces. For example, if I split all the pieces in 3/5 into three new pieces, there will be 9 pieces. And, instead of 5th parts, they will be 15th parts. If you have an image and you split even the "white pieces" into three new ones, you'll see those 15 parts. So, 3/5 = 9/15.

The arrow notation shown in the video has one arrow between the numerators and another between the denominators. It also has a little "x3" written next to it. This is to signify into how many pieces we split the existing pieces.

This notation can help students not confuse equivalent fractions with fraction multiplication. The two fractions are equivalent or the "same amount of pizza"; one is not three times the other.

Please also see this free sample worksheet: Equivalent Fractions worksheet. This worksheet shows the same notation and the same idea as the video. It is a sample from my book Math Mammoth Fractions 1.

Please let me know what you think of this notation. I haven't see it anywhere else, but maybe it does exist somewhere. Do you think it confuses or helps students?

## Comments

We used the arrow notation you demonstrated at Houghton--many moons ago--for pretty much the same reason as you mentioned.

I've posted an example here.

Love the video presentations! Keep up the good work!

I have had it happen to me where I would write something, and then later notice a similar thing in my old math book from when I was in school. (This wasn't with fractions though.) It's like those things reside somewhere within the deep recesses of the brain.

One thing that might help in the lesson is to explain why the splitting converts to multiplication. It's easy to see, once explained, but I'm not sure my students would "get it" just from the pictures: Each piece is split into three pieces, so we now have three times as many pieces, so the numerator has to be multiplied. At the same time, the new pieces are smaller (each is only 1/3 third the size it was), so the denominator--which tells the size of the pieces--is multiplied in the same way.

I've been following your blog for a little while now and I'd just like to say you have a lot of good ideas. I'm a HS Math teacher by trade.

I often use this kind of arrow notation to show that one quantity is "linked" to another in a variety of situations. I think it is kind of a natural way to show relationships and we use them because they are intuitive and easy for students to understand and follow. Arrows are familiar, where other mathematical notation is not.

You may have seen this in your past, but just as likely, you invented it on your own because it just feels right.

Yes, it is an interesting presentation. Let me just share with you how I taught my daughter (9 yrs) the “concept” of equivalent fractions.

Te kids wont understand (feel) certain of the statements like “no change of value” and etc. because they wont have the feel about the “value” as such. They would have a question within themselves that “what this value is”?.

In order to make them to feel the “value” and the phenomenon of “no change in value”, I used the method of cutting a round cake and a rectangular chocolate bar.

I cut the round cake in the same form you have explained and weighed it and showed the value as weight. In the second case, I broke the chocolate into 4 and 8 equal rectangular pieces and re-joined them to show the value in terms of size and area.

She enjoyed this learning.

Please feel free to call/mail me, the moment you feel I could be of some input.

Regards

Subrahmanya bk

Marketing Professional (Engineering)

India

Email: Aasha.Research@Gmail.Com

Phone: +91 (0) 94482 38383

Thank you

I was trying to teach my daughter about LCM (lowest common multiple) and HCF (highest common factor). Would you please suggest me a method, shoemting inline with the way you told about teaching equivalant fractions.

- Subrahmanya bk

So, you'd find LCM of 12 and 9, which is 36, and then convert both fractions so they have a denominator 36.

After you find the LCM, then you can think of the equivalent fractions like shown in the video.

Greatest common factor is typically used when simplifying fractions. For example, to simplify 36/96 you could find the GCF of 36 and 96 which is 12, and then simplify by dividing both the denominator and numerator by it.

However, it's not necessary to find the GCF in order to simplify. It is enough to find

anyfactor (such as 2), and simplify, and then continue simplifying until the denominator and numerator do not have any common factors. So a person could simplify in steps like this: 36/96 = 18/48 = 9/24 = 3/8 and arrive at the correct answer (simplified to the lowest terms).And it's not even necessary to use LCM when adding fractions. The least common multiple of the denominators is the most efficient, but you could use

anycommon multiple. It just may make your denominator to be a bigger number, and give you more work when simplifying the final answer.We are students of pedagogy in mathematics and computation, and we want to congratulate for the excellent blog that you created, cause the information appears there is used for a lot of people who have problems to understand mathematics in a fast form. Your blog was a big help for us because it served us to realize a better report for a subject of fundamental formation (English I). Finally, we want to thanks that exist people as you, who worry about the education not only in the class room, also in virtual world. Reiterating our congratulations and we hope that you continue with this excellent idea to explain in a easy and didactic form some fundamentals things in mathematics.

Atte. To You

Katherine Vega and Pablo Serrano

First Year Students of Pedagogy in Mathematics and Computation

University Catholic's Maule, Chile

We are students of pedagogy in mathematics and computation, and we want to congratulate for the excellent blog that you created, cause the information appears there is used for a lot of people who have problems to understand mathematics in a fast form. Your blog was a big help for us because it served us to realize a better report for a subject of fundamental formation (English I). Finally, we want to thanks that exist people as you, who worry about the education not only in the class room, also in virtual world. Reiterating our congratulations and we hope that you continue with this excellent idea to explain in a easy and didactic form some fundamentals things in mathematics.

Atte To You

Katherine Vega and Pablo Serrano

First Year Students of Pedagogy in Mathematics and Computation

University Catholic's Maule, Chile