Who wants to discuss teaching fractions?

[daijobu]

I never had the luxury of time to think about this while I was actually homeschooling, but now that I'm finished, I've been taking a deep dive into education pedagogy.  

Some people seem to argue that we should be using only the number line to define and illustrate fractions, while far more people are dividing up pizzas and rectangles into small pieces.  

Do you have a strong opinion about how to teach fractions?  

[Rosie_0801]

The most useful thing I found for fractions was the early lessons of CSMP using the minicomputers. Through them, dd became accustomed to breaking numbers into pieces, rearranging them and putting them back together. The idea that 1 could also be broken up, rearranged and put back together seemed perfectly natural by the time we got to it.

[EKS]

The most important thing is knowing what 1 is.

Then you should divide all the things.

[HomeAgain]

I teach fractions in a variety of ways.  First we do it orally and with blocks.  Right now I have a 2nd grader whose lessons consist of activities like. "can you double the four?  What do you have now? What if we take that eight and split it into two equal pieces?  What's half of eight?  Right!  That's one half of eight.  What do you think two halves of eight are?  How about three halves of eight?

Eventually I start writing it out, 1/2 of 8, 2/2 x 8, 3/2 x 8 , while I'm asking.  Then I just write.

That's the first introduction to fractions.  Working it into regular multiplication and division of numbers to understand it.  Then we start working with mixed fractions and improper fractions and everything else.  We move to Fraction Formula, a game of tubes and pieces where they can really compare and see that 3/12 is the same as 1/4, and develop a number sense of cutting into smaller pieces.

I've tried other methods over the years, including Math U See, which does have nice fraction pieces, and Gattegno, who introduces repeated ratio blocks, but going slow and introducing one new concept at a time with something that feels natural, like play or working with money or whatnot, it makes it less scary to see it on paper for a lot of kids I've worked with.

[Lori D.]

No idea about best formal pedagogy, but first exposure here was along about grade 2/3 with pattern blocks and a booklet that guided into making wholes with the different shapes of the pattern blocks, so you could see the whole was divided into pieces (fractional parts). For my very visual/concrete learner, that was an excellent intro to fractions.

Edited July 12 by Lori D.

[happypamama]

I use a lot of things. Some kids like a number line. Fraction circles or squares, with pieces, help some. Thinking about circles as pies really helps some, including my current almost 11yo. Pies work well when you're talking about fractions of a fraction. I do a LOT of "if you, sibling, other sibling, and other other sibling want to share this thing or things, how would we do that?"

 

I care less about current trends and more about speaks to the kids in front of me. When we talk about equivalent fractions, it made sense to them to think about multiplying by 3/3 as putting on a 3 shirt; it's still the same fraction, just with a different shirt. When we do borrowing, we have cookies on a plate. When someone wants more cookies, we open a box of 10, then a carton of 10 boxes, and so on until we have forklift loads on tractor-trailers. 

[Clarita]

3 hours ago, daijobu said:

Some people seem to argue that we should be using only the number line to define and illustrate fractions,

Curious what's the argument for this?

Even though my children have not officially learned fractions (as part of their math curriculum), it's too late because their parents without thinking it through already described reading analog clocks and dividing up circular food using fractions. 

 

[luuknam]

Also interested in the reasoning. I have a vague theory that maybe they *want* kids to focus on abstract reasoning, and/or are concerned that pizza makes understanding negative fractions or dividing by a fraction harder? Like, not many people would attempt to divide 15/4 pizzas among -1/3 of a person. 

I'm team pizza (etc) all the way. Who could possibly be against pizza?!? 🤣 ❤️🍕

Edited July 12 by luuknam

[HomeAgain]

7 hours ago, Clarita said:

Curious what's the argument for this?

Even though my children have not officially learned fractions (as part of their math curriculum), it's too late because their parents without thinking it through already described reading analog clocks and dividing up circular food using fractions. 

 

I'm curious, too.  My thoughts go toward the simplicity.  You don't need anything except pencil and paper and it's a constant form.  It makes it easy to illustrate, but I'm not sure if it's easier to understand.

The kids I work with tend to struggle with math anyway.  And I do try to keep things consistent so the manipulatives we do use are just a few, but these are kids that need to see what is going on in more than one way.  If I hand them a 1/2 tube and ask them "what's 1/3 of 1/2?"  They can do it, first by matching up pieces then by learning how it works on paper. But they need the feedback from something they can touch and make sure is right.

[Xahm]

My oldest is just going into algebra and my youngest into Kindergarten, so we're not anywhere close to done with math. For all of them, they learned basic fractions through real life before a math book, and they haven't found fractions hard. (The younger two have only gotten a smattering of fractions in formal math, if that much, but the older two have done more.) They probably got introduced to the written form of fractions by baking with me and looking for the 1/8 tsp and such. I told them early on that it has the 8 on the button because it means that 8 of those can equal one of the regular 1 tsp. I'm pretty sure each one tested that it at some point, but since they were toddlers when this started, I said it multiple times to each one until they were explaining it back to me before I had the chance. With 6 people in our family, we double lots of recipes, so they figured out early on that 2*1/4=1/2 and such. When they later saw it in math books, it wasn't any more surprising than whole number math. I can't see that only using a number line would have improved their in the least.

[SilverMoon]

Both? We started with pizza slices because it was easy to visualize.  It worked like a charm. As they became more proficient we picked up the number line. 

[Paige]

I'm doing fractions now and we do a lot of different things. My littlest is a very quick math student so I'm teaching him differently than the others. We used fraction manipulatives but we've been working with the fraction/division connection from the start. The fraction line indicates division, we connect it to decimals (money), and I show him how it divides out, and he also has used a number line divided into tenths showing decimals and fractions (separate lines). I have talked about how sometimes it's easier to think about a fraction because of the nice mental picture it shows and that decimals can be a pain to work with and it's a situational choice. His biggest problem is remembering what goes where on the line and mixing up the numerator and denominator.

I don't think a number line would be as useful for teaching equivalent fractions as little circle and rectangle pieces. I don't understand why teachers would be told to limit their options for teaching fractions by only using number lines. Kids can't hold a line and it's not even how they would experience fractions in the real world. Kids share pizzas with their families, they have piles of cars that they make groups out of, they might have 3 cookies and 4 friends and want to share it. 

[domestic_engineer]

@daijobu would you clarify what you meant with the term “number line”?  Are you referring to a simple line with tick marks that is often used to explain basic addition/subtraction?  Or are you referring to a rectangular model/manipulative similar to what Rightstart math uses?  https://store.rightstartmath.com/rightstart-fraction-magnet/
 

I’ve only used rightstart to teach fractions, and they use rectangular pieces as the main mental model rather than cutting circles/pizzas. (They cover fractions within a circular whole, but the instruction always go back to the rectangular pieces.). I believe their argument is that it’s easier for children to compare fractions, say 1/8 to 1/9, or 5/10 to 1/2, using rectangular pieces rather than pie pieces.   Once the foundation is set with rectangular pieces, I bring in the idea of dividing circles.   I think both rectangles and circles can be made concrete to kids. I refer to granola bars, toblerone bars, candy bars, pies, cakes, cookies, and pizzas.  🤪

Also, I think that multiplying a fraction by a fraction may be easier to model with rectangular wholes rather than circular wholes because the Area model can be utilized. (But to be fair, I’ve only taught their concept with rectangles and never with circles.)

ETA: If you meant a true, horizontal number line, how does one even make that meaningful to students?  I’d need an explanation of how they do that. 

Edited July 12 by domestic_engineer

[HomeAgain]

I have a set like this in my workroom: magnetic number lines for fractions. I tend to bring it out with Singapore math students because it's a lot like what they see in their books/makes the bar model consistent, but it's not something I ever begin a lesson with. 

[daijobu]

15 hours ago, Clarita said:

Curious what's the argument for this?

 

Thanks for asking!  🙂  I don't even recall how I taught fractions myself, just that I followed the Singapore Math curriculum and that was that.  But since homeschooling, I encountered Hung-Hsi Wu, an emeritus professor of mathematics (notably, NOT math education) at UC Berkeley.  He's written several papers and books on math education.  See:  https://www.amazon.com/Understanding-Numbers-Elementary-School-Mathematics/dp/0821852604/

He views elementary math education from the perspective of a mathematician.  In this article https://math.berkeley.edu/~wu/fractions1998.pdf he writes:  

"This approach to fractions is a geometric one, where a fraction is identified unambiguously as a point on the number line. Thus a fraction becomes a clearly defined mathematical object."

"To someone not familiar with the mathematics of elementary school, the fact that we give a clearcut definition of a fraction must seem utterly trivial. After all, how can we ask students to add and multiply and divide fractions if we don’t even tell them what fractions are? Unfortunately, it is the case that school texts usually do not define fractions."

"I have chosen to define a fraction as a point on the number line (while dodging the explanation of what a “number” is), and have introduced addition, multiplication, and division on fractions by extending the intuitive idea—gleaned from experiences with natural numbers—of what these operations ought to be."

"It is likely that teachers would expose students to other pictorial representations of fractions at this point, such as a square divided into 4 equal parts, or a pie cut into 6 equal parts, etc., so a caveat is in order. There is certainly no harm in introducing these models, but I would suggest doing so only after students have become proficient at working with the number line and the division of line segments. One reason is that our reasoning throughout the development of fractions is done with the help of the number line. But there is another reason: the pie representation of fractions, for instance, has the drawback of being clumsy at representing fractions > 1 because teachers and students alike balk at drawing many pies. Reasoning done with the pie model therefore tends to accentuate the importance of small fractions.  On the other hand, the number line automatically puts all fractions, big or small, on an equal footing so that they can all be treated in a uniform manner. An additional advantage is the flexibility of this model in all kinds of discussions, and this advantage would become most apparent when we come to the multiplication of fractions."

 

 

[daijobu]

38 minutes ago, domestic_engineer said:

@daijobu would you clarify what you meant with the term “number line”?  

ETA: If you meant a true, horizontal number line, how does one even make that meaningful to students?  I’d need an explanation of how they do that. 

 

You start with a horizontal line with arrowheads on both ends to indicate the line extends forever in both directions.  The integers consist of equally spaced points on that line, with zero in the center, positive integers increasing to right, negatives integers decreasing to the left.  

Now one might ask what about all the other points on the number line that are NOT integers?  What would we call those?  For example, what about the point that is directly right in between the zero and the one?  For integers we use 10 digits and place value, but what about these other points?  

Then we proceed with the ideas of the denominator and numerator, for example, we can have a denominator equal to three which means between each integer we mark off 3 equally sized parts, and then the numerator tells you at which third you are at starting from zero.  

[daijobu]

Checking Art of Problem Solving, Beast Academy seems to use the number line definition:

And PreAlgebra:

[daijobu]

Here's another sample from BA, Level 4, only number lines:

[EKS]

Note that neither the number line nor the pizza address this use of fractions:

There are 15 marbles, 5 of which are purple.  What fraction of the marbles is purple?

The key is identifying what 1 is and how it's being divided.

Edited July 13 by EKS

[daijobu]

7 minutes ago, EKS said:

Note that neither the number line nor the pizza address this use of fractions:

There are 15 marbles, 5 of which are purple.  What fraction of the marbles are purple?

The key is identifying what 1 is and how it's being divided.

That's a good point.  

[Clemsondana]

My own kids grasped fractions fairly quickly so I don't really remember what I did besides mostly following Singapore Math's program.  But, I often volunteer with kids working on fractions and different things seem to help different kids see what is happening.  When their assignment involves a number line, we often talk about it similarly to the pizza...we're looking to put 3/4 on the number line, so let's divide the distance between 0 and 1 into 4 parts.  If you have all 4 parts, you have one whole line.  What would have at the first mark?  1/4...etc.  And then if you have 5/4, that's more than 1 whole so we need to take the distance to the next number (2) and divide it into 4 parts too!  I'll also toss in that, while theoretically putting things on a number line (or any number of other ideas to teach abstract concepts) can help the kids to have a deeper understanding, they can also quickly become rote.  Which may be the point - the kids may internalize the concepts that way.  Once kids figure out what they are supposed to do for the number line, they quickly learn that they should make a line and put some whole numbers on it, then make divisions for whatever number is in the denominator and then start counting.  They aren't necessarily thinking 12/5 is 2 and 2/5 - they just make a line with numbers up to 5 and then start dividing it into 5ths and then count to 12.  I encourage them (and occasionally the assignment asks) to write 5/5, 10/5, etc over the whole numbers so that they can visualize it.  For helping my younger see while dividing by a fraction is the same as multiplying by the reciprocal, we worked that out with pizza slices ever year (kid could do it by rote but was adamant that it shouldn't work).  

[Clarita]

This is why you would introduce fractions using a number line vs. a pie. 

Note both Beast Academy and Hun-Hsi Wu assume that prior to this official introduction to the term and symbol of fractions young children had exposure naturally to things/shapes/amounts being split in half or quarters or whatever. Even in Beast Academy 1 they had children dividing shapes in to equal pieces and use the English fraction term for it like halves, or a fourth, etc.

5 hours ago, EKS said:

There are 15 marbles, 5 of which are purple.  What fraction of the marbles is purple?

Beast Academy 5 describes this as a ratio, "In some ways, ratios work a lot like fractions. So, we often use fraction notation to write and compare ratios. However, unlike fractions, ratios are not numbers on the number line. Ratios simply describe relationships. For example, we don’t add or subtract ratios."

[EKS]

8 hours ago, Clarita said:

Beast Academy 5 describes this as a ratio, "In some ways, ratios work a lot like fractions. So, we often use fraction notation to write and compare ratios. However, unlike fractions, ratios are not numbers on the number line. Ratios simply describe relationships. For example, we don’t add or subtract ratios."

I think that this is being unnecessarily pedantic and that insisting that the number line is the only way to represent fractions is being unnecessarily restrictive.

In the case of the marbles, because the ratio is comparing the relationship between a part and a whole, it works the same as a fraction.  In the context of the problem you may or may not add or subtract them.  Example:  There are 15 marbles, 5 of which are purple and 3 of which are red.  What fraction of the marbles is neither purple nor red?  

Once kids understand that these sorts of ratios work the same as fractions, then you can introduce the sort where you are comparing parts to parts.  Example: What is the ratio of red to purple marbles?

I think it's important to stress how things are the same in math whenever you can. It's also important to be able to use different mental models for the same concept as needed.

Edited July 13 by EKS

[almondbutterandjelly]

I did three main things with my visual, big picture learner.  1.  Fractions are just division problems (she had already learned how to divide).  2.  We constantly used Lakeshore Learning's set of fraction manipulatives.  https://www.lakeshorelearning.com/products/math/fractions-decimals-percents/building-fractions-activity-center/p/HH978/.  3.  Key to Fractions books.

[luuknam]

19 hours ago, Clarita said:

Note both Beast Academy and Hun-Hsi Wu assume that prior to this official introduction to the term and symbol of fractions young children had exposure naturally to things/shapes/amounts being split in half or quarters or whatever. Even in Beast Academy 1 they had children dividing shapes in to equal pieces and use the English fraction term for it like halves, or a fourth, etc.

Okay, I thought that OP meant introducing fractions on the number line before introducing them as dividing shapes into half and such, which just seemed beyond silly to me (and which is why I said I'm team pizza all the way (and by pizza I meant all physical shapes, definitely not just circles)). I did use the number line later on as well, but the kids' introductions to fractions (including the addition and subtraction of fractions) in preK-2 or so was with concrete objects, circular, rectangular, fraction towers, etc (not entirely sure when I added in the number line... could've still been during that time period or maybe later, but definitely *after* concrete objects).

Trying to introduce basic fractions to a preschooler with only a number line seems *really* odd to me, and withholding the concept of fractions until a substantially later age in order to do so also seems odd.

[daijobu]

I think timing is what was tripping me up.  Dr. Wu's ideas I think are geared toward grade 4-5, right before pre-algebra.  Prior to that, most students are learning unit fractions and not doing operations on them.    

[EKS]

9 minutes ago, daijobu said:

I think timing is what was tripping me up.  Dr. Wu's ideas I think are geared toward grade 4-5, right before pre-algebra.  Prior to that, most students are learning unit fractions and not doing operations on them.    

In thinking about this further, I now suspect that using the number line is setting the stage for an introduction to the real numbers, which are a way bigger deal that they are made out to be in K-12 math.

[prairiewindmomma]

We started fraction work when they were preschoolers….we needed 6 forks because there were six people in our family, and everyone got one fork—each person got one out of the six things. 
 

By the time Youngest was doling out stuff, Oldest was eating 3 out of the 16 slices of pizza. He would eat 2/8 slices or 1/4 of the pepperoni pizza and 1/8 of the cheese pizza.

I tried to purposefully use “divided by” as part of the language of showing notation because 5/4 is the same as 5 divided by 4…

2 of my kids are dyscalcic, so we have spent a LOT of time on place value. One of the other things I am careful to teach from very early on is the relationship between numbers….that subtraction is the inverse of addition, and so on. We spent a lot of time constructing and deconstructing numbers singapore math style, and we spent a lot of time on drawing representations of what we are doing (charts, graphs, etc) Singapore style because my dyscalcic kids have a hard time visualizing quantities. The leap to fully abstract math goes a bit slower for them. A couple of my kids have intuitively understood fractions, a couple of mine have needed substantial support to grasp. Once one of my kids can figure out the relationship between fractions, they are usually ready to tackle deeper and more abstract math concepts.

[Penderwink]

I think the key thing to remember about fractions is that it’s a way of writing “one divided by two” or “three divided by seven”, not this separate idea of a half, or three lots of a seventh (because who can really picture a seventh). I was doing well in elementary school math and all the typical fraction work, but I remember there was a point I realized this, and it completely changed how I pictured it in my head. There is so much focus on half cups in baking, and half-past on clocks, I think it’s easy to lose focus of what fractions really mean

A lot of fraction work is easier with this mental picture, and almost thinking of fractions as a way of making things easier without a calculator.