Fraction division part 4: our final post on this subject (for now)

Well, all good things must come to an end. In our previous three blog posts, we discussed some affordances of using diagrams to understand fraction division. In this post, we will talk about why it is important for students to go beyond diagrams.

The limits of diagrams for solving fraction division problems

In our earlier posts, we argued that diagrams can help students see the structure of a problem and understand why it can be represented by division. However, diagrams are rarely efficient for carrying out the resulting fraction division. For example:

Mateo filled a 1 pint measuring cup with water until it was $\frac{7}{16}$ full. If a recipe calls for $\frac23$ pints of water, what fraction of the recipe can Mateo make with the water in the measuring cup?

Drawing a diagram for this problem is not the most efficient method (try it!). A student who has learned to see this as $\frac{7}{16}\div \frac23$ (through working with diagrams) would most efficiently calculate that value using the invert-and-multiply rule without worrying about a diagram for that particular calculation.

Explaining the fraction division rule using algebra

Last time we argued that the “How much in one group” interpretation with the right kind of diagram can help us see why dividing by a fraction is the same as multiplying by its reciprocal.

For example, a diagram that represents a situation where $\frac25$ of a number is $1\frac34$ can show that we can multiply $1\frac34$ by $\frac52$ to find that number.

What if we were to think about this from a completely algebraic perspective? By the definition of division,$$1\frac34 \div \frac25 = x$$ means that: $$\frac25 x = 1 \frac34.$$

To solve an equation like this, we simply multiply both sides of the equation by the multiplicative inverse of $\frac25$:

$$\frac52 \cdot \frac25 x = \frac52 \cdot 1 \frac34.$$

In other words: $$ x =1 \frac34 \cdot \frac52.$$

We are not claiming that students need to be able to make a formal argument like this in order to justify the general rule for dividing fractions! But they do, eventually, need to be able to solve specific equations of the form $$\frac25 x = 1 \frac34.$$

Students who can solve equations flexibly might find the solution  by rewriting an unknown factor problem as a division problem: $$x = 1\frac34 \div \frac25,$$ or by multiplying both sides of the equation by the reciprocal of $\frac25$: $$\frac52 \cdot \frac25 x = \frac52 \cdot 1 \frac34.$$

Both methods were implicit in many of the fraction division problems students have been conceptualizing with the help of diagrams, although there may not have been an explicit equation in those problems.  Using equations formalizes, makes explicit, and encapsulates the implicit understandings. So students who investigate fraction division with diagrams should have the opportunity to make connections to algebraic approaches as well.

Final thoughts

Fraction division is a topic that students encounter at a key time in their transition from their work in elementary school arithmetic to their study of algebra as generalized arithmetic in middle school and beyond. Appropriate use of diagrams can help them understand how fraction division relates to their earlier study of division of whole numbers and when a problem can be represented by fraction division. Diagrams can also mediate students’ transition to a more structural, abstract understanding of fraction division that is represented using numeric and algebraic expressions and equations. In general, diagrams can play a key role in helping students make the transition from arithmetic to algebra, as we have illustrated in the particular case of fraction division.

Division of fractions part 3: why invert and multiply?

We ended the previous post with a bit of a cliffhanger, with two possible diagrams to represent $1\frac34 \div \frac12$:

The first of these diagrams is more familiar to students because it reflects their past work, but the second is more productive for understanding “dividing by a unit fraction is the same as multiplying by its reciprocal.”

Why is the first one more familiar? In grades 3 and 4, students study both the “how many in each (or one) group?” and “how many groups?” interpretations for division with whole numbers (see our last blog post for examples). In grade 5, they study dividing whole numbers by unit fractions and unit fractions by whole numbers. But, as we mentioned in that post, in grade 5 the “how many groups?” interpretation is easier when dividing whole numbers by unit fractions because students do not have to worry about fractions of a group. Going from $3 \div \frac12$ to $1\frac34 \div \frac12$ using this interpretation feels fairly natural:

The main intellectual work here is seeing that $\frac14$ cup is $\frac12$ of a container, but because the structure of the problem is the same and that structure can be easily seen in the diagrams, students can focus on that one new twist. The transition also helps students see that “how many groups” questions can be asked and answered when the numbers in the division are arbitrary fractions.

So the “how many groups” interpretation is useful for understanding important aspects of fraction division and has an important role in students’ learning trajectory. It enables students to see that dividing by $\frac12$ gives a result that is 2 times as great. But it doesn’t give much insight into why this should be the case when the dividend is not a whole number.

The “how much in each group” interpretation shows why. Here are diagrams using that interpretation showing $3 \div \frac12 = 2 \cdot 3$ and $1 \frac34 \div \frac12 = 2 \cdot 1 \frac34$.In fact, the structure of this context is so powerful, we can see why dividing any number by $\frac12$ would double that number: $$x \div \frac12 = 2 \cdot x = x \cdot \frac21$$

This is true for dividing by any unit fraction, for example $\frac15$:In the diagram above, we can see that $1\frac34$ is $\frac15$ of a container, so a full container is $1\frac34 \div \frac15$. Looking at the diagram, we can see why it must be that the full container is $5 \cdot 1 \frac34 = 1 \frac34 \cdot \frac51$.

With a little more work to make sense of it, we can use this interpretation to see why we multiply by the reciprocal when we divide by any fraction, for example $\frac25$:In the diagram above, we can see that $1\frac34$ is $\frac25$ of a container, so a full container is $1\frac34 \div \frac25$. We can see in the diagram that $\frac12$ of $1\frac34$ is $\frac15$ of the container, so our first step is to multiply by $\frac12$: $$1\frac34 \cdot \frac12$$

Now, just as before, to find the full container, we multiply by 5: 

$\left (1\frac34 \cdot \frac12 \right) \cdot 5 = 1\frac34 \cdot \frac52$

This shows that dividing by $\frac25$ is the same as multiplying by $\frac52$!

There is nothing special about these numbers, and a similar argument can be made for dividing any number by any fraction. Now students, instead of saying “ours is not to reason why, just invert and multiply,” can say “now I know the reason why, I’ll just invert and multiply.”

Next time: Beyond diagrams.

Fraction division part 2: Two interpretations of division

In our last post, we asked people if they could come up with a division story problem for $1\frac34 \div \frac12$. Interestingly, almost all of the responses used the “how many groups?” interpretation of division. When interpreting multiplication in terms of groups, the two factors play different roles, and so there is another interpretation of division worth exploring.

If we say that $a \times b$ means $a$ groups of $b$, then

  • a division situation where $b$ and $a\times b$ are known but $a$ is unknown is called a “how many groups?” division problem
  • a division situation where $a$ and $a\times b$ are known but $b$ is unknown is called a “how many (or how much) in each (or in one) group?” division problem.

[Pause here and see if you can come up with a “how much in one group?” story problem for $1 \frac34 \div \frac12$.]

How do these two interpretations of division come into play as students learn about fraction division? In grade 5, students solve problems like $6\div \frac12$ and $\frac12 \div 6$. What’s nice about problems involving a whole number divided by a unit fraction or a unit fraction divided by a whole number is that we can think of them using the same structure that we thought about division of whole numbers.

  1. Kiki has 6 kg of chocolate chips. How many 2 kg packets of chocolate chips can she make?
  2. Kiki has 6 kg of chocolate chips. How many $\frac12$ kg packets of chocolate chips can she make?

Notice that these are both a “how many groups?” division problem, and because there is always a whole number of unit fractions in 1, the solution will be a whole number of groups (so students do not have to worry about fractions of a group). If we write equations to represent these problems, that can also help us see the structure:

$$? \times 2 = 6$$

$$? \times \frac12 = 6$$

  1. Nero had 6 cupcakes and 3 friends he wanted to share them with equally. How many cupcakes does each friend get?
  2. Nero had $\frac12$ of a cupcake and 3 friends he wanted to share them with equally. How many cupcakes does each friend get?

Notice that these are both a “how many in each (or how much in one) group?” division problem, and students don’t have to worry about fractional groups because the whole number in the problem is the number of groups.

Again, with equations:

$$3 \times ? = 6$$

$$3 \times ? = \frac12$$

So in grade 5, students can build on their understanding of whole number division without having to grapple with fractional groups, so long as they understand both of these interpretations of division.

In grade 5, students also learn about fraction multiplication, so they do encounter fractions of a group, but they are not required to put these two understandings together until grade 6 when they extend their understanding of division to all fractions. This provides some scaffolding for students on their way to understanding division of fractions in general.

Let’s look at these two interpretations of division for $1\frac34 \div \frac12$.

  • “How many groups?” : I need $1\frac34$ cups flour, but I only have a $\frac12$ cup measure. How many times do I have to fill the $\frac12$ measure to get $1\frac34$ cups flour? (A version of this was suggested by two different people on our last post.)
  • “How much in one group?” : I have a container with $1\frac34$ cups flour. The container is $\frac12$ full. How much flour does the container have when it is full?

Here are two possible diagrams to represent these two interpretations of division:

Next time: how the different interpretations of division and diagrams can be used to understand the “invert and multiply rule” and other approaches to understanding this procedure.

Illustrative Mathematics 6–8 Math

I can’t help writing this off-cycle blog post to celebrate the release of Illustrative Mathematics 6–8 Math  last Friday, a proud achievement of the extraordinary team of teachers, mathematicians and educators at Illustrative Mathematics (IM), one that I didn’t dream of when I started IM almost 7 years ago with a vision of building a world where all learners know, use, and enjoy mathematics.

Conceived initially as a  project at the University of Arizona to illustrate the standards with carefully vetted tasks, IM has grown into a not-for-profit company with 25 brilliant and creative employees and a registered user base some 40,000+ strong. Our partnership with Open Up Resources (OUR) to develop curriculum started a little over 2 years ago when we submitted a pilot grade 7 unit on proportional relationships to the K–12 OER Collaborative, as OUR was then known. In the fall of 2015, not understanding that it couldn’t be done, we agreed to write complete grades 6–8 curriculum ready for pilot in the 2016–17 school year.

One of the things I love about the curriculum is the careful attention to coherent sequencing of tasks, lesson plans, and units. The unit on dividing fractions is an example, appropriate to mention in the middle of this series of blog posts with Kristin Umland on the same topic. It moves carefully through the meanings of division, to the diagrams that help understand that meaning, to the formula that ultimately enables students to dispense with the diagrams. It illustrative perfectly our balanced approach to concepts and fluency. Kristin and I will be talking about that more in the next few blog posts.



Fraction division part I: How do you know when it is division?

In her book Knowing and Teaching Elementary Mathematics, Liping Ma wrote about this question and how teachers responded to it:

Write a story problem for $1 ¾ \div ½$.

[Pause here and think about the answer yourself.]

Many people find it hard to come up with a story problem that represents fraction division (including many math teachers, engineers, and mathematicians). Why is this hard to do? For many people, their schema for dividing fractions consists almost entirely of the “invert and multiply” rule. But there is much more to thinking about fraction division than that. So much in fact, that we can’t say it all in a single blog post. This is the first of several musings about fraction division.

The trouble with English

Consider this problem:

If you have 12 liters of tea and a container holds 2 liters, how many containers can you fill?

You probably know instantly that this is a division problem and that the answer is 6, because you know your times tables, and specifically you know that $2 \times 6 = 12$. If we say that $a \times b$ means $a$ equal groups of $b$ things in group, then a division problem where $b$ and $a\times b$ are known but $a$ is unknown is called a “how many groups?” problem. Here are some other questions that ask “how many groups?”

  • If you have 1 ½ liters of tea and a container holds ¼ liter, how many containers can you fill?
  • If you have 1 ¼ liters of tea and a container holds ¾ liters, how many containers can you fill?
  • If you have ¾ liter of tea and a container holds 1 ¼ liters, how many containers can you fill?

Some people think that the last one feels like a trick question because you can’t even fill one completely. Because we know the answer is less than one, we could also ask it this way:

  • If you have ¾ liter of tea and a pitcher holds 1 ⅓ liters, how much of a container can you fill?

So a division problem that asks “how many groups?” is structurally the same as a division problem that asks about “how much of a group?”, but because of the way we speak about quantities greater than 1 and quantities less than 1, the language makes the structure harder to see.

What other ways might we see the parallel structure?


Equations: $$? \times2 = 12, \quad ? \times \frac14 = 1\frac12, \quad ? \times \frac34 = 1\frac14, \quad ? \times 1\frac14 = \frac34.$$  The diagrams don’t have the language problem. In all cases the upper and lower braces show the relation between the size of a container and the amount you have.  Whether a whole number of containers can be filled (diagrams 1 and 2), a container plus a part of a container can be filled (diagram 3), or only a part of a container can be filled (diagram 4), the underlying story is the same.

Many people think of diagrams primarily as tools to solve problems. But sometimes diagrams can help students see structure or reveal other important aspects of the mathematics. This is an example of looking for and making use of structure (MP7).

The equations have an even clearer structure, but more abstract. They all have the structure $$\mbox{(quantity of containers)}\times\mbox{(size of a container)} = \mbox{(how much you have)}.$$

The intertwining of the abstraction of the equations and the concreteness of the diagrams is a good example of MP2 (reason abstractly and quantitatively).

Coming up next week: what else are diagrams good for?

A world without order (of operations)

What would such a world look like? Like this:
(((3\times(x\times x)) – (7\times x)) + 2).
$$What a world it would be! A world without ambiguity! A world where PEMDAS would just be P! A world where they would have to relocate the parenthesis keys to a more convenient location on the keyboard!

Parentheses, and order of operations, tell us how to read the meaning of an expression, how to parse it, not what to do with it. In the expression above, every matched pair of parentheses contains something of the form $$
(\mbox{blob}) * (\mbox{another blob}), \qquad (\mbox{where * stands for $+$, $-$, or $\times$}),
$$ unless the blobs are just numbers or letters, in which case we don’t surround them with parentheses. We always know exactly what things we are adding, subtracting, or multiplying. Starting with the outermost parentheses, we see it contains the sum of 2 and a blob. Looking inside that blob we see that it contains a blob minus another blob. And so on. The structure of the expression can be represented in a diagram:

So what is order of operations about, and why do we need it? Well, that’s a lot of parentheses up there, so it is useful to have some conventions about when things are understood to be a blob, without actually putting in the grouping symbols (blobbing symbols?). First, any sequence of multiplications and divisions is understood to be a blob (that’s the precedence of multiplication and division over addition and subtraction). Second, in a sequence of additions and subtractions, or of multiplications and divisions, you read from left to right. (Actually, there is disagreement about this last one in the case of multiplication and division, but never mind.) The first rule allows us to write the expression above as
((3\times x\times x- 7\times x) + 2).
$$The second rule allows us to leave out all the remaining parentheses. And, of course, we have other conventions about representing multiplication by juxtaposition, and about exponent notation, which allow us to write
3x^2 – 7x + 2.

Calling it order of operations is problematic because it can be misconstrued as suggesting that there is a specific order in which you must perform operations. There isn’t, except insofar as you sometimes have to wait to perform an operation until you have calculated all the blobs in it. But, for example, there is no law that says you have to do the multiplications first in $101\times56-99\times56$ and, in fact, it is more efficient to factor out the $56$ and do a subtraction first. Order of operations tells us how to read this expression: it’s a difference of two products, not a product of three factors the middle one of which is a subtraction. But it doesn’t tell us how to compute it. The word “order” in “order of operations” is best understood as referring to order in the sense of hierarchy, as in the diagram above.

Outside of textbook school mathematics the order of operations is a matter of common law, not constitutional law, and it’s a bad idea to make a federal case out of it on assessments. For example, dinging a student for interpreting $x/2y$ as $x/(2y)$ rather than $(x/2)y$ would be unreasonable; many scientists would do the same thing. If there is any danger of ambiguity we should put the clarifying parentheses in.


A few final thoughts:

    • thanks to Brian Bickley for suggesting the topic for this post
    • there’s a nice discussion of the history of order of operations over at the Math Forum
    • and bonus question: do we have to give multiplication precedence over addition? Could we do it the other way around?

New look, new title

I have updated to a new WordPress theme, partly because I thought it was time for a makeover, and partly to see if it would cure some of the problems people have had commenting. In the process I decided to change the title of the blog. My recent writings have been about school mathematics generally, and I hope they are of use to teachers everywhere, whatever their standards. I will still write occasionally about the Common Core, and I will still answer questions over in the forums. I have changed the settings in the forums to allow anonymous posting for people who have trouble logging in. I may have to change that back again if it causes security problems. And, speaking of security, the site has been protected by SiteLock since last summer’s hacking, which means that you will occasionally encounter a captcha screen.

And, by the way, the url also points to this blog.

Truth and consequences: talking about solving equations

The language we use when we talk about solving equations can be a bit of a minefield. It seems obvious to talk about an equation such as $3x + 2 = x + 5$ as saying that $3x+2$ is equal to $x + 5$, and that’s probably a good place to start. But there is a hidden assumption in there that the equation is true. In the Illustrative Mathematics middle school curriculum coming out this month we start students out with hanger diagrams to represent such equations:

The fact that the hanger is balanced embodies the hidden assumption that the equation is true. It is helpful for explaining why you have to perform the same operation on each side when solving equations; if you take two triangles from the left side you have to take two triangles from the right side as well in order to preserve the balance. This leads to a discussion of how performing the same operation on each side of an equation preserves the truth of the equal sign.

But what happens with an equation like $3x + 2 = 3x + 5$? In this case, the hanger diagram is a physical impossibility: the right hand side will always be heavier than the left hand side. I can imagine that students who have an idea of an equation as “the left hand side is equal to the right hand side” might be confused by this situation, and think this is not a proper equation. Especially when they reduce it to $2 = 5$. Students learn to say that this means there are no solutions, but it’s hard to make sense of that response rule without understanding what’s really going on with equations.

The fact is, an equation with a variable in it is neither true nor false, because it is merely a phrase in a longer sentence, such as “If $3x + 2 = x + 5$ then $x = \frac32$.” This sentence is true, but the phrases within it are not sentences and have no inherent truth or falsity. When we perform the same operation on each side of an equation, we are not only preserving the truth of the equal sign but also preserving the consequences of the equal sign. If we use if-then language when talking about equations, then we can make sense of equations with no solutions. A sentence like “If $x$ is a number satisfying $3x + 2 = 3x + 5$ then $2 = 5$” makes perfect sense. It’s the mathematical equivalent of “If the moon is green cheese, then I’m a monkey’s uncle.” It’s a way of saying the moon is not green cheese . . . or that there is no solution to the equation.

The middle schooler’s version of if-then language might not always use the words “if” and “then.” You might say “Imagine there is a number $x$ such that $3x + 2 = x + 5$. What can you say about $x$?” Just as you say “Imagine this hanger is balanced and the green triangles weigh one gram. How much do the blue squares weigh?” I think it’s a useful approach with students to remember that equations are a matter not just of truth, but of truth and consequences.

Why is a negative times a negative a positive?

OK, I can hear the groans already. There are many contexts for answering this question and they are dubious in varying degrees because the real answer is “because I said so.” That is to say, the rule for multiplying negatives is a convention; adopted for good reasons, but a convention nonetheless. Those good reasons are mathematical: we want to make sure that when we extend multiplication and addition to negative numbers the properties of operations still apply. In particular, we want the distributive property to apply. Meditate on this:
3\cdot(5 + (-5)) = 3\cdot5 + 3 \cdot (-5).
The left side is really $3 \cdot 0$, so it had better be zero. So the right side had better be zero as well. The first term on the right side is 15, so the other term had better be $-15$. So $3 \cdot (-5) = -15$. We want the commutative law to hold, so we had better say $(-5)\cdot 3 = -15$ as well. Now meditate on
(-5)\cdot(3 + (-3)) = (-5)\cdot 3 + (-5)\cdot(-3).
The same reasoning tells us that $(-5)\cdot(-3) = 15$.

Trouble is, all this is really hard to explain to middle schoolers, so people invent contexts. One context I’ve seen has something to do with sending out bills. If you receive 5 bills for 3 dollars then you have $5 \cdot (-3) = -15$ dollars. Sending out is the opposite of receiving, so if you send out 5 bills for 3 dollars, you have $(-5)(-3)$ dollars. But once you receive payment, you have \$15. So $(-5)(-3) = 15$.

One problem with this is that you have to buy more conventions to believe it: the convention about negative amounts of money representing debt, the convention about negative receiving being kinda sorta like sending out. That’s a lot of conventions to prove something that is, as I said, a convention itself. Another problem is that all this context really shows is that $-(-3) = 3$, five times. The multiplication in this context is really just repeated addition; it doesn’t work for numbers that are not integers. You can’t send out 5.6 bills.

There is one context that I think does a better job here, and that is $\mbox{distance} = \mbox{speed} \times \mbox{time}$. This does work with non-integers, and you can make sense of all of the quantities involved as negative numbers. Let’s assume that an object is moving along the number line, and that you measure its position at different times, setting your stop watch to 0 when it passes through the origin. Negative distance is distance to the left; negative speed is speed from right to left; and negative time is time before you started measuring. (Later we use the terms displacement and velocity, but there’s no need to introduce them right away.)

So if the object is moving at $-5$ m/sec, where is it at time $-3$ seconds? Well, it’s moving from right to left and it has 3 seconds before it hits the origin, so it is 15 m to the right of the origin. So $(-5)(-3) = 15$.

Was I cheating there? Is this context subject to the same objections I made about the money context? Didn’t I just make up a whole bunch of conventions about negative distance, time, and speed? I think these conventions pass the cognitive sniff test better. They don’t seem as artificial to me. You can really make quantitative sense of negative distance, speed, and time. It feels more like the real world and less like an accountant’s convention. (No offense to accountants intended.) In a way, we have replaced the mathematician’s desire to have the properties of operations continue to hold with the physicist’s desire to have the laws of physics continue to hold.

So where is the distributive property in all of this? I think it is built into our physical intuition about this context. If I travel for 3 hours, and then for another 2 hours, I can figure out how far I have gone by just adding the times and multiplying by my speed, or I can add the distances traveled in each time period. That’s the distributive property. If you dig into the reasoning I gave for the object moving at $-5$ m/sec in the light of this common sense, questioning each claim, you end up with something not too far from the mathematical reasoning I gave earlier.

By the way, this is the approach we take in the Illustrative Mathematics middle school curriculum. Finding contexts for mathematical ideas that are faithful to the mathematics is difficult and requires real sensitivity to both the mathematics and the way students think. Our brilliant curriculum writing team is up to that challenge.

Talking about fractions, decimals, and numbers

When students first learn about fractions, we want them to learn that they are just numbers; new numbers, but numbers nonetheless, that fit into the same system as the whole numbers they are familiar with. The number line can help with this, with whole numbers and fractions sitting together, and located in essentially the same way; choose a unit (1, 1/3, 1/10) and then count off a number of those units. It also helps students understand that equivalent fractions are just different ways of writing the same number. When (finite) decimals come along, they get added to the list of representations.

The Common Core emphasizes this unity by treating decimals as just a different way of writing fractions, e.g. in 4.NF.C: “Understand decimal notation for fractions, and compare decimal fractions.” In this view, 0.3 is not a new sort of number, just a different way of writing the number 3/10.

This leads to some difficulties in the use of language, because at some points in the curriculum you do want to distinguish between decimals and fractions, for example when you ask a student to write 4/5 as a decimal or to write 0.125 as a fraction. (“You told me it’s already a fraction!” the smart student might reply.)

The IM curriculum writing team was talking about these difficulties the other day and Cathy Kessel had a useful comment:

There’s a developmental issue. When fractions are introduced, the distinction between number represented and representation is blurred, and similarly for decimals (finite, then repeating). But, when the two types of representations are seen as representing the same thing, then the thing and its representations start to separate more.

Because we want students to develop a conception of the number behind the representation, we start out saying decimals are also fractions. Later we build a negative addition to the number line and add the opposites of fractions. Once we have a robust conception of the number line, inhabited by rational numbers, we want to talk about different ways of expressing those numbers: fractions, decimals, infinite decimals, expressions involving square root symbols and exponents. So we start to distinguish between fractions and decimals, not as numbers, but as forms for expressing numbers. We initially suppress their role as forms in order to gain a robust conception of number; once they are firmly attached to that conception we can distinguish between them.

They only way to do this without giving multiple meanings to the same words would be to invent new words and be consistent in their use. This harks back to the distinction between “numeral” and “number” in the New Math, which didn’t take hold.