Whether or not your state is using Common Core, all state standards for the order of teaching fractions are fairly consistent. There are key places where standards for explaining fractions should build upon students’ prior understanding of concepts such as whole numbers, and prepare those students for the mathematical concepts they will encounter in the future.
In this on-demand webinar, Dean Ballard, Director of Mathematics at the Consortium on Reaching Excellence in Education (CORE), explores a few of his favorite strategies for explaining fractions.
In this webinar, you will learn:
Below are a few highlights from topics covered in this webinar.
In the sample snippet above, Ballard explores key concepts for explaining fractions including:
“One of my favorite things to do in math with kids is to guide students to look for and identify patterns, and to generalize from these patterns,” says Ballard, explaining that the ability to spot patterns is not only important to understanding fractions but to grasping more general concepts in mathematics.
Ballard shares ideas for activities for teaching fractions that help students search for and formulate patterns within the division of fractions. By inviting students to look for “shortcuts” instead of, for instance, counting on a number line each time they encounter a fraction, students begin to explore patterns, look for mathematical relationships, and create new rules and algorithms for themselves. Building on students’ own sense-making teaches both content and process together for true mathematical proficiency.
It is very important that students understand the mathematical idea that any quantity can be written or represented in a number of different ways. This seemingly straightforward concept is a foundational idea which students will continue to build upon as they move on to middle school, high school, and eventually into college mathematics.
By explaining fractions through a variety of concrete, visual, and abstract methods, students can begin understanding fractions, fractions’ connection to whole numbers, and the concept that all numbers and quantities can have multiple equivalent representations.
As Dean Ballard explains in this on-demand webinar, effectively teaching fractions will help students understand core mathematics principles that they will build upon for the rest of their education.
For more insight into explaining fractions as well as engaging and useful activities for teaching fractions, watch the full, on-demand webinar from CORE.
We’re going to look at connections among fraction concepts. The two concepts I want to take on here are building fractions from unit fractions and patterns in division of fractions. So building fractions from unit fractions. Well, unit fractions are a building block to both compose and decompose fractions.
Recall that unit fractions are fractions with the numerator of one, therefore, whatever the denominator is… And recall, as I said before, the denominator tells us the size of the unit, the fraction, represents one of these units. The concept of building quantities from a single unit began with whole numbers where we build from the ones unit.
Now, suppose we have an eighth. So there’s an eighth. By using iteration, I can create more copies of this eighth and we can build three-eighths. Well, with fractions, we want students to see, that it’s important for them to see, that three-eighths is actually three one-eighths. And students that understand this, they’re able to decompose fractions into pieces that represent the same total, because now they can see, oh, I could do three and those eighths, plus an eighth, plus an eighth. And I could actually write it different. It’s the same as an eighth plus two-eighths.
Now, this is really important because this transitions to fractions that are greater than one. So suppose we have eleven-eighths. Now we can say, “Oh, well, I could decompose that into eight-eighths and three-eighths.” And that would be the same as one plus three-eighths, so that’s why eleven-eighths is he same as one in three-eighths.
Now, patterns with division. This is the other topic I wanted to get. So that was units and how units play an important role. And now I want to talk about patterns with division. I have to say, one of my favorite things to do with math with kids is to guide students to look for and identify patterns, and to generalize from these patterns. I’m not just talking about fractions. I’m just talking about in general.
And if you know your standards for mathematical practice then you’ll recognize this as part of the standard seven and eight. Now, division with fractions, surprising to most people, actually lends itself well to this idea of looking for patterns till they get to the algorithm. A beginning approach really is to move students toward the standard fractions division algorithm, invert multiply, as you know it, is to use what students learned previously about dividing with some basic fractions to identify patterns so that it can be generalized.
So a look there. First chart there, the table on the left where we dividing three by three. Well, three by three, three divide by three’s one. Three divided by one is three. Now, three divided by a half. Well, gain, I could go back to the number line and count how many halves are in three. Those six of them. Three divided by a third. I can count how many thirds will be in a three. They’ll be nine of them. How many fours in a three, there’s 12. How many fifths in a three, again, I can count those on a number line, see there’s 15 of them and how many fourths in a four. There’s 16.
Now, I can ask students to look for a pattern in the answers. Is there a shortcut we can find to doing this without having to count them on a number line each time. And hopefully, and with some guidance, we see that, hey, I can multiply the dividend, that whole number, by the denominator. Yeah, by the denominator of the divisor.
So if I multiply the whole number times the denominator, I get the answer. So they can write a rule for that. Whole number times a denominator gave me the answer. So there’s just a very beginning rule to start with, just a first step. Then we look at that second, the middle table, and we start again. So now we’re going to do three divided by fourth. We saw that was 12 already. Now, three divided by two-fourths. Again, I can go to a number line and count how many two-fourths are in three and see there are six of them.
And with three divided by three-fourths, I can go and count how many three-fourths are in three and see there’s four of them. I can do it again with six and see, okay, six divided by one-fourth. There’s 24 of those. By two-fourths, there’s 12 of those. By three-fourths, there’s eight of those. And again I asked it, look for a pattern here. Let’s build off that pattern we saw in the first table where we multiplied that dividend, that whole number, times the denominator. Then what happens?
And we start to see, oh, when it was one-fourth, we divided by, it was 12. And it was two-fourths, it was six. And then it was three-fourths, it was four. You start to see that relationship that we’re taking that answer and dividing it by whatever the numerator is. So we get a new rule. Whole number times a denominator, divided by the numerator gets us now another step closer.
Look at that third table. Three divided by one-fourth. Again, that’s 12. Now we’re going to keep the one-fourth the same, the divisors are the same. You keep changing the dividend. Three over two divided by one-fourth. Again, I can look at that on a number line and identify and count exactly how many fourths are in three-halves. Three fourths divided by a fourth. On a number line, I can count that. There’s three of them. Three-twelfths divided by one-fourth. There’s one of them.
Now, we look for a pattern here. And again, building off of that first pattern we saw. We take that three, which in each case is there. Multiply it times denominator, gets us 12. And then dividing that 12 by whatever the denominator of the dividend is. So students can write this as a rule. First numerator times the second denominator divided by the first denominator.
Well, we’re not quite there yet. But students actually can already now solve this problem. What is four-fifths divided by one-eighth. And the key here is that we start to put this together. We can put those last two tables together and work with kids to get to the algorithm, therefore all fractions, we just invert and multiply as a nice shortcut to get there. But they’ve already down most of the work here, seen the pattern, and made some sense of it.
It’s worth mentioning that materials today also provide some mathematical arguments and properties and diagrams to help show how the fraction division algorithm can be derived through properties using both partitive and measurement division. These methods, they’re pretty complex, in my view, and I’ve found them difficult for both teachers and students to make sense of. However, they do provide a valid mathematically structured justification for the invert multiply algorithm.
I’m not arguing against these or other approaches. I’m just showing what is it like about the patterns approach. Not that it’s the only approach to use, not the only approach I use. I mean, it’s not the only thing I do with the division fractions. And it’s a nice to have students to begin to build to this algorithm by them identifying patterns to the mathematics students creating the mathematical rules for the patterns they see, and by guiding students to generalize this math.
Certainly, there’s still plenty of teacher guidance that is still important to bring the point home and to [inaudible] what students notice and describe into a final division algorithm. But I love how it’s built on sense-making students do with the mathematics along the way. And this is instruction, in my mind, that teachers both content and process together for real mathematical proficiency.