I’ve been meaning to let you know about the Illustrative Mathematics blog, which launched a few weeks ago. It has blog posts by members of the IM community about our grades 6–8 curriculum and about teaching practice, including a whole series on the 5 practices framework of Smith and Stein. Also, we will be cross posting any IM related posts I write here over there as well. I hope you find our new blog useful!

# Why is the graph of a linear function a straight line?

In my last post I wrote about the following standard, and mentioned that I could write a whole blog post about the first comma.

8.F.A.3. Interpret the equation $y = mx + b$ as defining a linear function, whose graph is a straight line; give examples of functions that are not linear.

For example, the function $A = s^2$ giving the area of a square as a function of its side length is not linear because its graph contains the points $(1,1)$, $(2,4)$ and $(3,9)$, which are not on astraight line.

The comma indicates that the clause “whose graph is a straight line” is nonessential for identifying the noun phrase “linear function.” It turns the clause into an extra piece of information: “and by the way, did you know that the graph of a linear function is a straight line?” This fact is often presented as obvious; after all, if you draw the graph or produce it using a graphing utility, it certainly looks like a straight line.

When I’ve asked prospective teachers why this is so, I’ve gotten answers that look something like this:

We know that a linear function has a constant rate of change, $m$. If you go across by 1 on the graph you always go up by $m$, like this:

So the graph is like a staircase. It always goes up in steps of the same size, so it’s a straight line.

This is fine as far as it goes. It identifies the defining property of a linear function—that it has a constant rate of change—and relates that property to a geometric feature of the graph. But it’s a “Here, Look!” proof. In the end it is showing *that* something is true rather than showing *why* it is true. Which is to say that it’s not a proof.

Still, the move to a geometric property of linear functions is a move in the right direction, because it focuses our minds on the essential concept. We all know that any two points lie on a line, but three points might not. What is it about three points on the graph of a linear function that implies they must lie on a straight line?

Because a linear function has a constant rate of change, the slope between any two of the three points $A$, $B$, and $C$ is the same. So $|BP|/|AP| = |CQ|/|AQ|$, which means there is a scale factor $k =|AQ|/|AP| = |CQ|/|BP|$ so that a dilation with center $A$ and scale factor $k$ takes $P$ to $Q$, and take the vertical line segment $BP$ to a vertical line segment based at $Q$ with the same length as $CQ$. Which means it must take $B$ to $C$.

But (drumroll) this means that there is a dilation with center $A$ that takes $B$ to $C$. Dilations always take points on a ray from the center to other points on the same ray. So $A$, $B$, and $C$ lie on the same line.

I don’t really expect students to get all of this, at least not right away. I’d be happy if they understood that there is a geometric fact at play here; that seeing is not always believing.

# How do you tell if a function is linear?

Over at the IM Virtual Math Coach we got a question about the following grade 8 standard:

8.F.A.3. Interpret the equation $y = mx + b$ as defining a linear function, whose graph is a straight line; give examples of functions that are not linear.

For example, the function $A = s^2$ giving the area of a square as a function of its side length is not linear because its graph contains the points $(1,1)$, $(2,4)$ and $(3,9)$, which are not on astraight line.

I could write a whole blog post about that comma in the first sentence, but for now I want to focus on the question of what exactly are the student expectations entailed by this standard. Certainly students should be able to recognize that $y = mx + b$ defines a linear function; and they should be able to show a function is not linear by finding points on the graph with different slopes between them. (By definition, a linear function is one with a constant rate of change, that is, a function where the slope between any two points on its graph is always the same.) However, the following PARCC released item suggests the possible expectation that students be able to tell if a function is linear or not purely from looking at its defining equation.

My guess is that the writer of this item was thinking that students would detect non-linear functions by noticing the functions with squared and cubed terms (they could also use the point method but that doesn’t seem likely). That’s not a good path to lead them down. Sure, $y = 5-x^2$ is not linear. What about $y = 5 – x^2 + (x+1)^2$? By the look of it it is even more not linear! But of course it is in fact linear, because the expression on the right is equivalent to $2x + 6$. The grade 8 standards don’t expect students to be making such simplifications as expanding squares of binomials.

This is another example of the confusion between expressions and functions. The expression $(x+1)^2 – x^2$ has non-linear terms in it, but the function it defines is linear because it is equivalent to $2x + 1$. Equivalent expressions define the same function.

There’s another confusion revealed in this item, the confusion between equations and functions. Look at option C. Is it intended to be a distractor? Will a student who chooses it be marked wrong? Such a student would have a case for protest on the grounds that $-3x + 2y = 4$ is not a linear function because it is not a function, it is an equation. Certainly if you choose to think of $x$ as the input and solve for $y$ to get the output you can think of it as a function, which would indeed be linear. You could also go the other way around and choose $y$ as the input and get a different linear function. It is conventional when $x$s and $y$s are floating around to think of $x$ as the input and $y$ as the output, but you can flout convention without being mathematically incorrect.

The moral of this story is, I suppose, that it is easier to tell when a function is not linear than to tell when it is linear. Testing for non-linearity involves just picking a few points on the graph; testing for linearity involves picking every possible pair of points on the graph and verifying that the slope between them is always the same. It’s instructive to do this with $y = mx +b$: if $(x_1,y_1)$ and $(x_2,y_2)$ are two different ordered pairs satisfying this equation, then $$\frac{y_2-y_1}{x_2-x_1} = \frac{{mx_2 + b} – (mx_1 +b)}{x_2-x_1} = \frac{m(x_2-x_1)}{x_2-x_1} = m.$$

OK, next week I’ll write a blog post about the comma.

# The image of function in school mathematics

Well, oops, took a little break from blogging there. But I’m back now.

In the course of working on an article with the same title as this blog post for a publication about Felix Klein, I did a Google Image search on the word “function,” with the following results.

I find this fascinating for a number of reason. First, notice the proliferation of representations: graphs, tables, formulas, input-output machines, and arrow diagrams (those blobs in the second row). A corresponding search in a French, German, or Japanese gives a very different result. Here is the Japanese one (where I searched google.co.jp for “関数”).

This has a much greater focus on graphs, and no arrow diagrams. The French and German ones are even more focused on graphs; a search in Spanish, however, gives results similar to English.

What does all this mean? Well, I’ll be discussing this more in the article I’m writing, but for now I have just a couple of observations.

First, the English language search reveals a preoccupation with examples of relations that are not functions, with examples of graphs, arrow diagrams, and tables. I have always found this preoccupation perverse: the examples are artificial, and there is very little to be gained from them except an ability to answer questions about them on tests. This preoccupation is not evident in the searches in other languages.

Second, take a look at these two representations, from the English and Japanese language searches:

.

Together they represent a case study in designing representations. Can you see what’s wrong with the one on the top? A function is supposed to have only one output for every input. Does an apple slicing machine have that property?* The representation on the bottom, on the other hand, clearly represents how to think of addition as a function with two inputs.

I’m interested to hear readers’ thoughts on the representations that come up in these and other searches. Maybe someone who can navigate Baidu can tell us what the Chinese results are.

*To be fair, the authors of the web page with the apple slicing machine are clearly aware of the problem. But their contortions to get around it only emphasize the fundamental flaws in the representation.

# Math is not linear, but time is

A short post today with a question for our readers.

A number of years ago there was a popular piece by Alison Blank titled Math is not linear, which gave a number of ideas about the order in which we teach mathematics. A curriculum writer has to grapple with the fact that, although math is not linear, time is. Hermione Granger’s time turner does not actually exist. Tuesday comes after Monday, and Tuesday’s lesson comes after Monday’s lesson and, in the end, a teacher has to decide what to teach on each day; that is, they have to decide on a linear order in which to teach mathematics. The gist of “Math is not linear” is that that order need not be a dry march through a logical hierarchy of topics. You can, as Blank says, go on tangents, foreshadow topics to come, connect back to previous topics, and give students problems that create a need for a new topic. These are all great ideas.

Our question is: what other ideas do people have to make sure that the sequence of lessons in a course makes sense to students and makes sense mathematically? Do you recommend any books or articles that might help answer these questions? We have some ideas and will be writing some posts about them, but want to hear from the community as well. Please feel free to share your thoughts in the comments, or on Twitter with @IllustrateMath, #timeislinear.

# 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.

- Kiki has 6 kg of chocolate chips. How many 2 kg packets of chocolate chips can she make?
- 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:

$$? \times 2 = 6$$

$$? \times \frac12 = 6$$

- Nero had 6 cupcakes and 3 friends he wanted to share them with equally. How many cupcakes does each friend get?
- 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?

Diagrams:

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?