Category Archives: Questions from middle school teachers

QMST VI: Cross-multiplying

A dear friend asks (slightly edited),

Do you think there is any way to do cross multiplying with meaning?  I remember discussing this with you last year and I know you will probably say “no”.  But if we want to have students solve problems using any methods available to them, why not make this a method (not THE method but A method) available to them (discussing common denominators)? Honestly, if you were going to solve 12/123 = x/768,392.614, would you “scale up” or cross multiply?  I’m not trying to sound critical – I’m just about to teach this stuff and just trying to pick your brain.

That “with meaning” part is key for me. Consider the example presented here: 12/123=x/768,392.614. First let’s consider why we’re solving this proportion. I’m struggling to come up with a good one due to the great difference in magnitudes across these two fractions. So let’s just say they’re similar triangles. The numerators are the short sides of these triangles in centimeters; the denominators are the long sides (also in centimeters).

Cross-multiplying gives the equation 9,220,711.368=123x. What is the meaning of 9,220,711.368 in terms of our triangles? Well, it’s the product of the short side of the small triangle and the long side of the large triangle. I suppose we could think of it as the area of a rectangle with these two side lengths. But why should that area be the same as the area of the rectangle formed by the short side of the large triangle and the long side of the small triangle? And is this a new theorem?

The yellow triangles are similar, so the blue rectangle has the same area as the red one.

In any case, contrast that with the meaning involved in how I would really solve this proportion (and yes, I would really do it this way). 768,392.614÷123=6247.094… Now 12•6247.094=74,965.13

Let’s not argue about significant digits or rounding; those are tangential issues that could be raised about the originally proposed proportion. We can hash those out on someone else’s blog.

No, I want to talk about the meaning of that 6247.094. It’s the scale factor. It tells me how many times bigger the large triangle is than the small one.

If you end up with a student who can talk about the meaning of that 9,220,711.368, then by all means have her cross multiply. But in my classroom, I’m gonna insist on meaning all the way through. We don’t have to think about meaning at all times, but we have to be able to think about it.

QMST V: Creating an inviting classroom

Today I’m turning over the blog to a fabulous middle school math teacher who comes from an English teaching background. All those smart, smart questions from middle schoolers over the last 6 months? Her kids asked those. Give her a hand in thinking something through, would you please?

Hi.

I’m Alex Otto and I teach 7th and 8th grade math at Kodiak Middle School in Kodiak, Alaska.

I was looking for some feedback from my peers.  So much of what math teachers talk about is how to teach abstract concepts, yet so much of what is important to middle school kids is what is going on in their immediate surroundings.

Along those lines, I’m thinking about how to best arrange my classroom in terms of seating, bulletin boards, groups, “stations”, etc.  Does anyone have any ideas and especially links to pictures of “inviting” classrooms that facilitate the kind of learning that we want to take place?  (I feel like a lot of the “inviting” classrooms in my school are language arts classrooms with reading nooks, writing stations, etc. )

Questions from middle school teachers: Blog etiquette

The following from an email conversation with a long-distance teacher friend:

I was wondering if you actually have met everyone who writes on your blog?  Or are some of them people who just found your blog and wrote to you?

No I have not. I look forward to someday meeting some folks, but definitely do not know them all. And I read and comment on the work of folks whom I have never met in person. It’s like a parallel universe to which I am a relative newcomer. I find it fascinating.

What is the appropriate math-teacher etiquette?  I don’t know if it is a “closed” community of people who all know each other (like if everyone who posts on your blog is a math professor and all of you go to conferences together and know each other really well).

No, no and no. Of course it is natural to feel more comfortable jumping in to a conversation when you know the participants personally, but there is no requirement or expectation at all. All are welcome and encouraged to participate. And of course, the places where that conversation is explicitly encouraged and nurtured will have higher rates of participation (viz. Dan Meyer’s blog).

I have spent precisely two hours in the physical presence of Dan Meyer (who is a grad student, not a professor), three or four with Karim Ani of Mathalicious (who is a teacher and entrepreneur, not a professor) and zero total hours with all of the other folks whose work I read regularly (very few of whom are professors). If you’re a fan of The Office, you’ll identify with my claim that There is no inner circle. Really.

Are professors who run blogs usually open to hearing from middle school teachers (people they don’t know I mean)?

Absolutely. That’s why we write. We hope to have an audience and we hope to hear from and learn from our readers. (And again, the whole ‘professor’ thing? It’s just not so.)

Finally, from another source (a smart and talented former student of mine, now a high school teacher in North Dakota), let’s dispel this misconception:

And for no other reason than my spying on your blog, I decided to drop you a line

If it’s on the blog, you’re not spying. You’re reading. If it’s only on my hard drive? That’s spying.

And consider this your invitation to jump into the conversation.

Questions from middle school teachers: Division of fractions

The following was sent along to me by a dear friend and mentor. It is an excerpt from a conversation she was having with a seventh-grade teacher she works with:

What are your thoughts on teaching students to divide fractions with common denominators?

[With respect to the invert-and-multiply algorithm] I know that I have walked students through this process with fraction circles. Given the right practice problems using the reciprocal makes a lot of sense. It also gives us an opportunity to talk about reciprocals!

I just wonder if, since students appear to struggle immensely with fractions in general, that having a common process for the 4 operations would make it easier to find more success. Students who easily see patterns, with a good dose of number sense, would then discover the reciprocal piece on their own.

The first thing that struck me on reading this was meaning. As math teachers, we pay very careful attention to abstract representations and we sometimes allow the meaning to slip away.

Notice that the problem (learning to divide fractions) and the proposed solution (an algorithm) are both phrased in abstract terms. Neither mentions the meanings of the division process. A quick tutorial on meanings of division, then I’ll apply to fraction division.

sharing

Write a story problem that leads to whole-number division. Most adults given that task will produce something equivalent to:

I have 35 apples to share equally among 7 people. How many apples does each person get?

In this problem, we have some number of objects (35) that we are dividing up into a known number of groups (7). We need to figure out how many are in each group. This first meaning for division is variously known as sharing, partitive and how many groups? division. These terms all mean the same thing.

measuring

But there is another meaning for division. Consider the following problem.

I have 35 apples to package in bags. I can fit 7 apples in each bog. How many bags can I fill?

In this case, we know how many are in each group, but we don’t know how many groups we can make. This meaning for division is referred to as measuring, measurement, quotative and how many in each group? division. These terms all mean the same thing.

Application to fraction division

While the sharing interpretation of division is the iconic one for most people (adults included), it is harder to conceptualize in the world of fraction division than the measuring interpretation is. Consider the following two problems:

  1. I have 3/4 lb. of cookie dough. I am making large cookies using 1/4 lb. of dough. How many cookies can I make?
  2. I have 3/4 lb. of cookie dough. I am making large cookies and that 3/4 lb. is enough to make 1/4 of a cookie. How much do I need to make a whole cookie?

The first problem is the measuring problem. How many 1/4’s are in 3/4? Each “group” is of size 1/4 lb. I need to know how many groups I can make.

The second problem is the sharing problem. What is 3/4 one-fourth of? If 3/4 is 1/4 of a group, how much is in one whole group?

With enough exploration of these kinds of contexts, the common denominator algorithm for dividing fractions can emerge. If we are asking how many 2/3 are in 7/9, we can notice that it would be easier to think of 2/3 as 6/9. Then we can see that we have one group of 6/9 in 7/9, with 1/9 left over. That 1/9 is 1/6 of a group, so 7/9 ÷ 6/9 = 1 and 1/6.

And ultimately, 7/9 ÷ 6/9 just refers to 7 things and we are making groups of 6 things. The common denominators ensure that the things are the same as each other-they are each ninths of the same whole. So 7/9 ÷ 6/9 is equivalent to 7÷6.

The invert-and-multiply algorithm is more commonly associated with sharing situations.

There have been several good articles on fraction division in the journal Mathematics Teaching in the Middle School in recent years. The best of them (in my opinion) is titled “Measurement and Fair-Sharing Models for Dividing Fractions” by Gregg and Gregg. But a quick search on the MTMS website will turn up several more.

Summary

So, what do I think of teaching students the common denominator algorithm for dividing fractions? I want to start with meaningful situations for doing the division. If we do that, then the common denominator algorithm is the one that stands a chance of making sense.

If, by contrast, we are working on a purely abstract level, then I don’t much care which algorithm we use. Both are equally efficient (less canceling in the quotient with common denominators, but more time spent looking for common denominators). Both extend to algebraic fractions, which is what mathematicians are usually concerned with. Indeed, I can teach (and have taught!) an entire calculus course in which the only algorithm for division I use is common denominators.

But I would be wary of the surface similarities implicit in wanting to “hav[e] a common process for the 4 operations”. The process isn’t really the same. Notice that in the common denominator algorithm for dividing fractions, the common denominator disappears in the quotient, while it does not do so when we add or subtract. And using common denominators for multiplying fractions seems like a lot of wasted effort, since we’ll need to cancel out those extra factors at the end of the computation.

I say go for meaning. Then the commonalities will be within each operation-division always means sharing or measuring whether we are using whole numbers or fractions-rather than across the operations on an abstract level.

Wump hats, part II

In the previous post, I described a workshop I was leading with seventh grade teachers recently. We were working on a problem in which the Wumps are given hats. The first hat is described by points in the coordinate plane. The other hats are transformations of the original hat and the problem has students investigate the effects on the image of multiplying and of adding and subtracting.

I had 6 small groups of teachers and there are 5 image hats. I had each small group use gridded chart paper and markers to draw the original hat and one assigned image hat on the same coordinate axes, as below.

Wump hat 1

Wump hat 1

I asked the sixth group to make up their own rule and to draw the original hat and the image, but not to show the rule on their chart paper.

We put the drawings on the classroom wall and gathered to talk about them. I asked the teachers to match the hats with the known rules, which went well. The group came to consensus about which was which and observed informally that:

  1. adding and subtracting moves the original figure around, and
  2. multiplying changes the size of the figure

I then asked teachers individually to think about what rule might have created the sixth image (see below).

What rule generated this image?

What rule generated this image?

Based on the rules above, I expected that most of the teachers would create a rule such as (2×-1, y+3). This is based on the two observations above, but applied in the opposite order. The most obvious change from the original to the image is that the image is twice as wide, hence the “2x”. The other change is that the hat moved. The left corner of the image is 1 unit to the left of the original’s corner, and the image is 3 units higher. This accounts for the “-1″ and the “+3″.

I then asked the group that made the hat to reveal their rule. Much to my teachers’ surprise, the rule was (2×-2,y+3).

2x-2, not 2x-1

This is what my friend Jim had been asking me about over the years. The reason 2×-2 is correct is that observation number 2 above is not complete. Multiplying changes the size of the original, and by necessity it moves the points around. Consider this: if two points define the left and right bottom corners of the Wump hat, and if the Wump hat doubles in size, then those endpoints cannot possibly both be in the same place. For then they would be the same distance apart and the hat could not have changed size.

So multiplying changes the size, but it also moves the points around. The hat moves to the right as a result of the “2x” part of the rule. So we need to subtract 2 (not 1) in order to shift the hat to its final resting place.

I felt that I had created a wonderful moment where teachers were ready to learn some mathematics. They had made a prediction and they had been presented with evidence that these predictions were incorrect. In resolving such conflicts, we have opportunities to learn.

A close look at the hat in question will reveal that there is something unexpected about the diagram. The scales on the two axes are not equal. This became an important point in the ensuing conversation and I’ll examine that in the next post.

Wump hats, part I

The thing I enjoy most about working with practicing teachers is seeing them start to think about familiar mathematics in completely new ways. Bringing this about can be quite challenging in a room full of secondary-trained math teachers like myself. Because we are licensed to teach high school, we have studied a lot of undergraduate mathematics and we tend to value solutions according to the sophistication of the mathematical tools employed. It was hard for me to learn early in my teaching career that it would really be better if I knew a variety of unsophisticated strategies rather than a single sophisticated one. I was good at thinking like an undergraduate math major. I needed to learn to think like my students-not just like one of my students, but like any of my students. Learning to do this requires seeing familiar mathematics in a new light.

I was working on this in a professional development setting with seventh-grade teachers recently. Connected Mathematics (or the Connected Mathematics Project, CMP) is a National Science Foundation-funded middle school mathematics curriculum. The seventh-grade unit Stretching and Shrinking has student studying similarity and the relationships between similar figures-corresponding angles are congruent, while side lengths are proportional, for instance.

Stretching and Shrinking looks at similarity relationships as transformations. That is, if we have two similar figures, we think of one as the original and the other as the image. If the context suggests that one of these ought to be the original, we follow that. Otherwise, they may be arbitrarily chosen. For example, if we make a drawing of the car, the car is the original and the drawing is the image. If we make a drawing of a bacterium in a microscope, the bacterium is the original and the drawing is the image. As these two examples illustrate, the image may be smaller than the original, or it may be larger than the original.

In cases where original and image are not naturally defined, as with two similar triangles in a textbook, it is legitimate to consider either figure the original and the other the image. As students become familiar with the ideas of the unit, they tend to be more strategic about their choices; some prefer to have the smaller figure always be the original, while some prefer to have the figure with the most known side lengths be the original, for example.

The fundamental mathematical idea of the unit is scale factor. The scale factor between two figures is the number that the lengths of the original figure are multiplied by to make the lengths of the image. In the bacterium example, the image is larger than the original so the scale factor is greater than 1 (on a microscope, the scale factor is marked as the degree of magnification). In the car example, the image is smaller than the original so the scale factor from the car to the drawing is a fraction smaller than 1.

One way students are introduced to similar figures in the unit is through the coordinate plane. They draw a group of characters called the Wumps. The Wumps are charismatic little fellows (see below) who are defined by a small set of points in the coordinate plane. Mug Wump is the original Wump and students are provided a set of rules for transforming Mug into other Wumps. The trick is that some of these are not really Wumps, for to be a Wump you have to look exactly like Mug. You may be larger or smaller than Mug, but you must look just like him. So the rule (2x,2y) transforms Mug into Zug, who is a Wump. Zug is twice as wide and twice as tall as Mug and looks just like him. The rule (3x, y), though, transforms Mug into Lug, who is not a Wump. Lug is three times as wide as Mug and equally tall. Lug doesn’t really look like Mug at all; he is an impostor.

All of this is preliminary to an interesting experience I recently had while working with a group of seventh-grade CMP teachers. There are two parts to the experience. I’ll write about the first here, and the second part next week.

Once we have drawn the Wumps, we help them to get dressed. We start with hats. Like the Wumps, the Wump hats are defined by points on a coordinate grid, but fewer of them. What is new in the Wump hats problem is that the rules for transforming hats include addition and subtraction, not just multiplication. From the Wump problem, students know that similar figures have congruent corresponding angles and that rules of the form (ax, ay) will give images with side lengths a times as long as the original lengths. In the Wump hats problem, students consider what the effect will be of a rule like (x+1, y-2).

In many seventh grade classrooms, students come to the following two conclusions:

  1. adding and subtracting moves the original figure around, and
  2. multiplying changes the size of the figure

In my experience in professional development settings, teachers are usually happy to leave it at that as well. My dear friend Jim Mamer has asked me several times over the years about these conclusions, because they don’t tell the whole story. In particular, multiplying not only changes the size of the figure but also moves it. This is by necessity; if the points stay put, then the sides of the figure cannot get longer. I have never thought much about Jim’s questions on this because issues have not come up in my sessions.

But I made an instructional decision this time that brought the issue to the forefront. Continued in the next post.