Some of the things I want to highlight in this section, that I want for us to look at how these concepts connect from whole numbers to fractions, are the concepts of units, and also how fractions are numbers, and equivalent fractions, addition and subtraction with fractions, and the multiplication and division with the fractions.
Now, here’s a list from the common core state standards, standards related to this. And it shows fraction standards from each grade level in abbreviated form here. And then [inaudible] state’s not common CORE. These links and the order of the content to be developed with fractions over the years is pretty consistent in all state standards. And these standards here provide a nice direction as to some of the places where it makes sense to build fraction concepts on the foundation of whole number concepts.
On this chart, we can see several places highlighted in red or like orange, I think, that are prime places for connecting fractions with whole numbers, including places where the common CORE standards actually explicitly state that work with fractions should be linked to and built on prior understandings with whole numbers.
I’ve also noticed that in purple they are highlighted… There’s a yearly direction in the standards to use visual models. I just wanted to emphasize that since that’s what we just emphasized here in the webinar. All right. So that first thing, units. What about units? Well, with place value, we have all whole number units based on the ones unit. That’s the central unit value. Tens or 10 ones bundled together. Hundreds are 10 tens, or 100 ones bundled together, etc.
Tenths, hundredths, and thousandths, etc., should also be seen in relationship to ones, as part of the ones unit. Fractions are units as well. Halves, thirds, fourths, are all parts of a ones unit. The denominator… I’m sorry. The denominator tells us the unit. If the denominator happens to be a 10 or 100, or 1000, for instance, then the fractions easily converts to a decimal representation. But either way, the denominator is the unit and consistently part of a ones unit.
And this is the idea, that a ones unit is partitioned into smaller units. And by the way, any fraction with a one in the numerator is actually called a unit fraction. And for this reason, because the fraction is exactly one unit of whatever the denominator is. As I mentioned before, I’ll just repeat again, how important number lines are, because you need to understand that fractions are numbers, that you can count them just like you can with whole numbers.
The work we do on number lines, putting them on number lines, really keeps reinforced in the idea that fractions are numbers. They’re quantities, just like whole numbers are. Now, equivalent fractions. I want to talk about that here in a couple of different ways. First I just want to point out the idea of how we make equivalent fractions. And this is built on whole numbers. And I just first remind you of the multiplicative identity property.
Now, you may not remember the name of it, but the idea, everybody I think is pretty clear on, that any number times one equals the same number. So 18 times one equals 18. This is a very simple property, but it’s actually a very, very important property with not only whole numbers, fractions, but throughout algebra. And that’s throughout middle school and high school, and actually in the college level of mathematics. This very simple property plays a huge role all the way through.
Now, with fractions, this property works the same, really. So two-fifths times one. That still equals two-fifths. However, the end result doesn’t always look the same with fractions. That’s kind of the answer with whole numbers, of course. We multiply a whole number by one, it looks just like the same whole number in the end. But with fractions, it really usually looks different.
Suppose we multiply two-fifths by three-thirds. The result looks like a different value. I mean, it is a different fraction, but it’s not a different value. Then that’s the difference here. And this is where we need to help students. We need to help them really focus on the concept of multiplying by one, that the value of the quantity stays the same. The value stays the same. Using visuals such as that paper folding, looking at a number line, and using what a fourth-grade teacher I know calls multiplying by the big bad one. The result is another fraction that is equal in value to the first.
So just like with our paper folding, when two-fourths, and four-eighths are the same value as one-half of the sheet of paper, two-fifths times a one in the version of three-thirds equals six-fifteenths. It’s the same value. So this type of illustration and phrasing with big bad one and the connections to other concrete and visual representations help students really hold on to, remember, and make sense of this mathematics.
All right. Another thing about equivalent representations that I want to talk about is connecting this idea of equivalent fractions to this really big idea in mathematics, that there are multiple equivalent representations of numbers. Doesn’t matter what kind of numbers those are, but there are always multiple different ways to write it.
Students can build from recalling even with whole numbers, such as 451, that that can be expressed in many ways as you see just four of them here. And looking at that fourth one, D, there, we could see it’s expressed as three hundreds, 14 tens, and 11 ones, which is actually a necessary way to represent 451 because what if we wanted to subtract 273 from it, then that’s how we’re going to represent it because we’re going to need to regroup the hundreds and the tens and the ones to do the subtraction.
Likewise with fractions, they can be represented in multiple ways as we can see here. But if we needed to do something like subtract two-fifteenths from two-fifths, then it helps to represent two-fifths differently as really six-fifteenths. And so rerunning values that are different equivalent forms, that a big idea in math. And from operations, whole numbers, to fractions, and well beyond, as I mentioned before. Therefore it’s really important to make a big deal of this big idea in math, and facilitate students seeing how the idea that any quantity can be represented in multiple ways translates from whole numbers to fractions, and that is a general idea, not a specific procedure or skill that just happens with fractions.