### 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?

### Comments

J.D.Fisher said…
Interesting presentation, Maria.

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!
Maria Miller said…
Thanks for showing me this! It's good to know that the notation has been used somewhere else, too. I just wonder if I have seen it myself years and years ago somewhere, because I honestly can't tell if I have or not.

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.
Anonymous said…
I have also used that arrow notation, not only for equivalent fractions, but for just about anything that reduces to a proportion---ratio, percents, etc. I'm not sure where I learned it, or if it was something I invented independently.

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.
Beckie Russell said…
Hi Maria,

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.
"AuntEM" said…
This method is great! I can't wait to use it with my students. I have done a similar strategy but without the important notation shown in the video. It makes sense and i am sure that students of all learning styles will be able to better understand this concept. Thanks for sharing! I will post again later after trying it with the kids.
Anonymous said…
When I use the "traditional" method that you demonstrated, it helps the students realize that they are multiplying by one (3/3 = 1). One times any number is that same number, so we didn't change the value...just the size of the pieces.
Anonymous said…
Hi Maria
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
Unknown said…
Thank you very much for the lesson, it was very good and very clear, for me and for my son that is in grade 3, as here in doha Qatar, we do not have a lot of American teachers out of the school, also please be more confidetnt because you look and sound scared or shocked, also work a little bit better on the resolution of the image
Thank you
Anonymous said…
The video seemed fine here; so, it might be on your receiving end that the resolution was less than it appeared to you. It was, I assume, meant for the teacher to see for the strategy and not for use with the kids. Obviously, you got the technique and that is the important part. Many videos appear less than perfect on the Internet; but, if the content is understood it is up to you (us) to go from there in real-life with our students. I think the video was great!
Maria Miller said…
I upload the videos to YouTube in much higher resolution, but YouTube converts them to lower resolution so that the file size is not too large. So, I have not much control over it.
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Subbu said…
Hello Maria

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
Maria Miller said…
Least common multiple is normally used when you add unlike fractions. For example, 1/12 + 5/9. You need to find a common denominator and the most efficient one to use is the least common multiple of the two denominators.

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 any factor (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 any common multiple. It just may make your denominator to be a bigger number, and give you more work when simplifying the final answer.
Katherine Vega and Pablo Serrano said…
Ms. Maria Miller

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
Pablo said…
Ms Maria Miller

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
Mary Maggie said…
Hello! Is there any chance that you post your videos in TeacherTube (or SchoolTube)? YouTube is blocked in our public school system. The videos are great, and I think my students would benefit from them. :)
Maria Miller said…
I have tried TeacherTube and their site does not work... I cannot upload, they don't answer support emails, etc. but I've never heard of SchoolTube before. Will look into it.