I’m excited to share the 4th installment of the Making Sense Series which explores meaning, equivalence, and comparison of fractions.

Fractions are the gatekeeper of algebraic thinking and probably a big reason why we suffer from arithmophobia as a society. I’m hoping this progression helps provide some relief and courage moving forward. Let’s make sense of fractions together.

Happy viewing and stay thirsty.

Multiplicative identity language: quantity of one.

Some folks call it a magic one, the publishers of CPM Core Connections use it.

I love using the quantity of one for my students as it ties nicely as they progress to higher levels of math like polynomial division.

I like using the phrase copycat one because it gives my students a clear picture of the numerator and denominator being the same and a quantity of one.

I wrote more about it here: http://mathformiddles.com/fraction-rules/

Thanks for this awesome addition to your progression videos! I share these with my teacher learners who are taking the RESA math endorsement course, and it’s been really helpful for them to see how fractional concepts build upon one another!

Why did you stop at 4th grade? What about operating with fractions?

This is great! What a wonderful way to express this concept. One of the reasons that you are my favorite son-in-law!

This is wonderful. I am rewriting our math curriculum K-4, and your video did an excellent job of vertically aligning and summing up fractions. I added it to our curriculum as a teacher’s resource. Thank You!

Poking around on the internet I found this, “parts of”. An alternative view, fixing the names well before “fractions”:

http://www.themathpage.com/arith/parts-of-numbers_1.htm

I think you will find it interesting.

Thanks for these awesome videos!

You are very gifted at communicating complicated ideas into a simple, understandable way. I think your progression videos are the best I’ve seen in explaining and connecting these concepts, and I use them often to share with the teachers I work with. Love what you do sir, thank you for making us all better.

Thanks Jeremiah. I stand on the shoulders of giants. I’ve picked lots of brains in my teaching career, so the fact that it’s easy to understand is a compliment to the friends that have allowed me to pick apart their understanding along the way. All of us are smarter than one of us.

Great quote- “All of us are smarter than one of us.”

You are a mathematical rock star!

Graham,

This is such an awesome resource for teachers and parents too! Thank you so much for the time you put into making it. It’s awesome. AWESOME!

One phrase I try to avoid with my students is that “whatever you do to the numerator, you do to the denominator.” I find that rule expires once students start operating with fractions. It’s certainly not the worst phrase, but I still find myself avoiding it. Instead I try to link their understanding to the multiplicative identity (2/3 * 1 = 2/3 and 2/3 * 3/3 = 6/9 so 2/3 = 6/9).

In graduate school, I wrote a blog post on strategies for comparing fractions to help some of my classmates who only knew the butterfly method. At the risk of over-self-promotion, I figured I would share that here because it might be helpful for others who aren’t familiar with these strategies: http://www.mathfireworks.com/2015/04/5-strategies-for-comparing-fractions/

One of my favorite things about teaching is designing questions that “break” certain methods. For example, when students only know how to compare fractions with common denominators or with the butterfly method, I ask them to compare 433/2,346 with 433/297,862. Of course, the butterfly method still works here, but these problems are wonderful headaches to motivate students to search for a new aspirin.

Again, thank you for sharing these videos. Did I mention that they are awesome!? Please keep making them!

-Tyler

I second the feedback about ““whatever you do to the numerator, you do to the denominator.”

I would really love to have better language around this. It would be great to hear other people’s ideas.

Perhaps there is some mileage from the analogy with subtraction. For example, students learn and use strategies like:

33 – 15 = (33 + 5) – (15 +5) = 38 – 20

but we don’t have “33 – 15 = (33 * 5) – (15 * 3)”

Joshua & Tyler,

The “what we do to the denominator, we do to the numerator” always feels awkward and forced to me. The issue is that it’s pushed from the onset as a means to identify equivalence. It’s the elephant in the room because we all know it’s there. It exists. But we are extremely limited in how we can explain it.

Since posting I’ve gone back, watched the video and I agree that there has to be a better way (a more mathematical way) to describe this but I can’t wrap my head around it.

Still chewing.

I say something like, “If you multiply the denominator by [#], you have to multiply the numerator also, because if you make [#] times as many pieces, now [#] times as many of them are [selected/blue/whatever].”

Or: “If you divide the denominator by [#], you have to divide the numerator also, because if you make the pieces [#] times bigger, now the number of pieces and the number that are [selected/blue/whatever] each need to be divided by [#].”

Solid post on comparing fractions Tyler. Thanks for linking it in here and don’t worry about self-promotion. I have the power of the delete button in the comment section:-) It’s spot on and only adds to and compliments what’s already here! Thanks for the kind words bud. All of us are smarter than one of us.

Thanks Graham. Another great video for educators!!! Keep it up!!

Love this, Graham. Many thanks.

My favorite fraction article for preservice teachers has been Tad Watanabe’s TCM article on representation. These models are so complex, we don’t fathom a lot of what is going on as kids experience them. http://www.nctm.org/Publications/teaching-children-mathematics/2002/Vol8/Issue8/Representations-in-Teaching-and-Learning-Fractions/

Thanks John and I really appreciate you sharing the link as well. Tad’s work is unbelievable and has greatly influenced a lot of my fractional understanding.

We’re super thankful to have such knowledgeable people working with our pre-service teachers by pushing the “right” type of Kool-aid.

Thank you! This is yet another great resource allowing our teachers to look at the vertical alignment in preparing for our upcoming fractions units!

Thank you for making these. They are excellent.

Truly, Thank you.