Month: August 2017

Illustrative Mathematics 6–8 Math

/ Bill McCallum / 7 Comments

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?

/ Bill McCallum and Kristin Umland / 8 Comments

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?

A world without order (of operations)

/ Bill McCallum / 8 Comments

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?